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<meta name="description" content="Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. We should avoid variables when forming statements, as they may change the logical value.
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\(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\).
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$2=7$ statement $x=5$ not a statement In logic we do not use the equals sign, we use the equivalence sign $\equiv$.
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Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc."/>
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Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc."/>
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@ -32,7 +32,7 @@
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<meta name="twitter:description" content="Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. We should avoid variables when forming statements, as they may change the logical value.
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\(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\).
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$2=7$ statement $x=5$ not a statement In logic we do not use the equals sign, we use the equivalence sign $\equiv$.
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Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc." />
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<meta property="og:description" content="Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. We should avoid variables when forming statements, as they may change the logical value.
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<meta property="og:description" content="Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. We should avoid variables when forming statements, as they may change the logical value.
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\(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\).
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$2=7$ statement $x=5$ not a statement In logic we do not use the equals sign, we use the equivalence sign $\equiv$.
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Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc." />
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@ -151,7 +151,9 @@
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<div class="post-content">
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<div class="post-content">
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<ul>
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<ul>
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<li>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.</li>
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<li>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.</li>
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</ul>
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</ul>
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@ -163,11 +165,11 @@
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<li><p>We should avoid variables when forming statements, as they may change the logical value.</p>
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<li><p>We should avoid variables when forming statements, as they may change the logical value.</p>
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<ul>
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<ul>
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<li><span class="math">\(2=7\)</span> statement</li>
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<li>$2=7$ statement</li>
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<li><span class="math">\(x=5\)</span> not a statement</li>
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<li>$x=5$ not a statement</li>
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</ul></li>
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</ul></li>
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<li><p>In logic we do not use the equals sign, we use the equivalence sign <span class="math">\(\equiv\)</span>.</p></li>
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<li><p>In logic we do not use the equals sign, we use the equivalence sign $\equiv$.</p></li>
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<li><p>Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).</p></li>
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<li><p>Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).</p></li>
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@ -180,147 +182,147 @@
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<li><p>The operations done on propositional variables are called propositional connectives.</p>
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<li><p>The operations done on propositional variables are called propositional connectives.</p>
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<ul>
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<ul>
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<li>Conjunction: <span class="math">\(p \land q\)</span> is only true if both p and q are true <span class="math">\((0001)\)</span></li>
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<li>Conjunction: $p \land q$ is only true if both p and q are true $(0001)$</li>
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<li>Disjunction: <span class="math">\(p \lor q\)</span> is only false if both p and q are false <span class="math">\((0111)\)</span></li>
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<li>Disjunction: $p \lor q$ is only false if both p and q are false $(0111)$</li>
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<li>Implication (material conditional): <span class="math">\(p \implies q\)</span> is false only if p is true and q is false (truth table <span class="math">\((1011)\)</span>)
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<li>Implication (material conditional): $p \implies q$ is false only if p is true and q is false (truth table $(1011)$)
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<ul>
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<ul>
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<li><span class="math">\(\equiv \neg p \lor q\)</span></li>
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<li>$\equiv \neg p \lor q$</li>
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</ul></li>
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</ul></li>
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</ul></li>
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</ul></li>
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<li><p>Not necessarily connectives but unary operations:</p>
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<li><p>Not necessarily connectives but unary operations:</p>
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<ul>
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<ul>
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<li>Negation: Denoted by ~, <span class="math">\(\neg\)</span> or NOT, negates the one input <span class="math">\((10)\)</span>.</li>
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<li>Negation: Denoted by ~, $\neg$ or NOT, negates the one input $(10)$.</li>
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</ul></li>
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</ul></li>
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<li><p>A (propositional) formula is a "properly constructed" logical expression.</p>
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<li><p>A (propositional) formula is a “properly constructed” logical expression.</p>
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<ul>
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<ul>
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<li>e.g. <span class="math">\(\neg[(p \lor q)] \land r\)</span></li>
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<li>e.g. $\neg[(p \lor q)] \land r$</li>
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<li><span class="math">\((p \land)\)</span> is not a formula, as <span class="math">\(\land\)</span> requires 2 variables.</li>
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<li>$(p \land)$ is not a formula, as $\land$ requires 2 variables.</li>
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<li>Logical equivalence: <span class="math">\(\phi(p, q, k) \equiv \psi(p, q, k)\)</span>, logical value of <span class="math">\(\phi\)</span> is equal to logical value of <span class="math">\(\psi\)</span>.</li>
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<li>Logical equivalence: $\phi(p, q, k) \equiv \psi(p, q, k)$, logical value of $\phi$ is equal to logical value of $\psi$.</li>
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<li>Commutativity: <span class="math">\(p \land q \equiv q \land p\)</span></li>
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<li>Commutativity: $p \land q \equiv q \land p$</li>
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<li>Associativity: <span class="math">\((p \land q) \land r \equiv p \land (q \land r)\)</span></li>
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<li>Associativity: $(p \land q) \land r \equiv p \land (q \land r)$</li>
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<li>Distributivity: <span class="math">\(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\)</span></li>
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<li>Distributivity: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$</li>
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<li>Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
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<li>Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
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<ul>
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<ul>
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<li><span class="math">\(\neg(B \lor C)\)</span> can be written as <span class="math">\(\neg B \land \neg C\)</span></li>
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<li>$\neg(B \lor C)$ can be written as $\neg B \land \neg C$</li>
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</ul></li>
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</ul></li>
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</ul></li>
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</ul></li>
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<li><p>Double negation law: <span class="math">\(\neg(\neg p) \equiv p\)</span></p></li>
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<li><p>Double negation law: $\neg(\neg p) \equiv p$</p></li>
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<li><p>De Morgan's laws: <span class="math">\(\neg(p \land q) \equiv \neg p \lor \neg q\)</span> and <span class="math">\(\neg(p \lor q) \equiv \neg p \land \neg q\)</span>.</p></li>
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<li><p>De Morgan’s laws: $\neg(p \land q) \equiv \neg p \lor \neg q$ and $\neg(p \lor q) \equiv \neg p \land \neg q$.</p></li>
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<li><p>If and only if (<em>iff</em>): <span class="math">\(p \iff p \equiv (p \implies q) \land (q \implies p)\)</span></p></li>
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<li><p>If and only if (<em>iff</em>): $p \iff p \equiv (p \implies q) \land (q \implies p)$</p></li>
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<li><p>Contraposition law:</p>
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<li><p>Contraposition law:</p>
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<ul>
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<ul>
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<li><span class="math">\((p \implies q) \equiv (\neg q \implies \neg p)\)</span> prove by contraposition
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<li>$(p \implies q) \equiv (\neg q \implies \neg p)$ prove by contraposition
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<ul>
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<ul>
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<li><span class="math">\((p \implies q) \equiv (\neg p \lor q)\)</span></li>
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<li>$(p \implies q) \equiv (\neg p \lor q)$</li>
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<li><span class="math">\((\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)\)</span></li>
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<li>$(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$</li>
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</ul></li>
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</ul></li>
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</ul></li>
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</ul></li>
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<li><p>Contradiction law:</p>
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<li><p>Contradiction law:</p>
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<ul>
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<ul>
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<li><span class="math">\(p \lor \neg p \equiv 1\)</span> and <span class="math">\(p \land \neg p \equiv 0\)</span></li>
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<li>$p \lor \neg p \equiv 1$ and $p \land \neg p \equiv 0$</li>
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</ul></li>
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</ul></li>
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<li><p>Tautology: <span class="math">\(\phi (p, q, ... r)\)</span> is a tautology <em>iff</em> <span class="math">\(\phi \equiv 1\)</span></p></li>
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<li><p>Tautology: $\phi (p, q, … r)$ is a tautology <em>iff</em> $\phi \equiv 1$</p></li>
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</ul>
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</ul>
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<h2 id="sets">Sets</h2>
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<h2 id="sets">Sets</h2>
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<ul>
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<ul>
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<li><p>We will consider subsets of universal set <span class="math">\(\mathbb X\)</span></p>
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<li><p>We will consider subsets of universal set $\mathbb X$</p>
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<ul>
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<ul>
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<li><span class="math">\(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\)</span></li>
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<li>$2^\mathbb X = { A : A \subseteq \mathbb X}$</li>
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<li><span class="math">\(2^\mathbb X = P(\mathbb X)\)</span></li>
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<li>$2^\mathbb X = P(\mathbb X)$</li>
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<li>All 2 object subsets of <span class="math">\(\mathbb X\)</span>: <span class="math">\(P_2(\mathbb X)\)</span></li>
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<li>All 2 object subsets of $\mathbb X$: $P_2(\mathbb X)$</li>
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</ul></li>
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</ul></li>
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<li><p><span class="math">\(A \subset B \equiv\)</span> every element of A is an element of B <span class="math">\(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)</span></p></li>
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<li><p>$A \subset B \equiv$ every element of A is an element of B $\equiv {x \in \mathbb X : x \in A \implies x \in B}$</p></li>
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<li><p>Operations on sets:</p>
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<li><p>Operations on sets:</p>
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<ul>
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<ul>
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<li>Union - <span class="math">\(\cup\)</span> - <span class="math">\(A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}\)</span></li>
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<li>Union - $\cup$ - $A \cup B = { x \in \mathbb X : x \in A \lor x \in B }$</li>
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<li>Intersection - <span class="math">\(\cap\)</span> - <span class="math">\(A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}\)</span></li>
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<li>Intersection - $\cap$ - $A \cap B = { x \in \mathbb X : x \in A \land x \in B }$</li>
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<li>Complement - <span class="math">\(A'\)</span> - <span class="math">\(A' = \{ x \in \mathbb X : \neg (x \in A) \}\)</span>
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<li>Complement - $A’$ - $A’ = { x \in \mathbb X : \neg (x \in A) }$
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<ul>
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<ul>
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<li>If <span class="math">\(x = \{ 1 \}\)</span> then <span class="math">\(x' = \emptyset\)</span></li>
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<li>If $x = { 1 }$ then $x’ = \emptyset$</li>
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</ul></li>
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</ul></li>
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</ul></li>
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</ul></li>
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<li><p>Equality of sets: <span class="math">\(A = B\)</span> iff <span class="math">\(x \in \mathbb X : (x \in A \iff x \in B)\)</span></p></li>
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<li><p>Equality of sets: $A = B$ iff $x \in \mathbb X : (x \in A \iff x \in B)$</p></li>
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<li><p>Difference of sets:</p>
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<li><p>Difference of sets:</p>
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<ul>
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<ul>
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<li><span class="math">\(A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'\)</span></li>
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<li>$A \setminus B = { x \in \mathbb X : x \in A \land x \notin B } = A \cap B’$</li>
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<li>Symmetric difference: <span class="math">\(A \div B = (A \setminus B) \cup (B \setminus A)\)</span></li>
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<li>Symmetric difference: $A \div B = (A \setminus B) \cup (B \setminus A)$</li>
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</ul></li>
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</ul></li>
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<li><p>Laws of set algebra:</p>
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<li><p>Laws of set algebra:</p>
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<ul>
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<ul>
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<li><span class="math">\(A \cup B = B \cup A , A \cap B = B \cap A\)</span></li>
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<li>$A \cup B = B \cup A , A \cap B = B \cap A$</li>
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<li><span class="math">\((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\)</span></li>
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<li>$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$</li>
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<li><span class="math">\((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)</span> vice versa</li>
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<li>$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ vice versa</li>
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<li><span class="math">\(A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X\)</span></li>
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<li>$A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$</li>
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<li><span class="math">\((A \cup B)' = A' \cap B'\)</span> vice versa</li>
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<li>$(A \cup B)’ = A’ \cap B’$ vice versa</li>
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<li><span class="math">\(A \cup A' = \mathbb X, A \cap A' = \emptyset\)</span></li>
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<li>$A \cup A’ = \mathbb X, A \cap A’ = \emptyset$</li>
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</ul></li>
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</ul></li>
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<li><p>Note: <span class="math">\(\{ \emptyset \} \neq \emptyset\)</span>, one is a set with one element, one is the empty set, no elements (<span class="math">\(\{ \}\)</span>)</p></li>
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<li><p>Note: ${ \emptyset } \neq \emptyset$, one is a set with one element, one is the empty set, no elements (${ }$)</p></li>
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<li><p>Quip: <span class="math">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</span></p></li>
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<li><p>Quip: ${ x \in \mathbb R : x^2 = -1} = \emptyset$</p></li>
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</ul>
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</ul>
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<h2 id="quantifiers">Quantifiers</h2>
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<h2 id="quantifiers">Quantifiers</h2>
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<ul>
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<ul>
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<li><span class="math">\(\phi\)</span> - prepositional function: yields only true or false value</li>
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<li>$\phi$ - prepositional function: yields only true or false value</li>
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<li><span class="math">\(\forall\)</span> means "for all" and <span class="math">\(\exists\)</span> means "there exists"</li>
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<li>$\forall$ means “for all” and $\exists$ means “there exists”</li>
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<li><p><span class="math">\(\forall\)</span>:</p>
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<li><p>$\forall$:</p>
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<ul>
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<ul>
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<li>Shorthand for <span class="math">\(\land\)</span> e.g. <span class="math">\((\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0\)</span></li>
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<li>Shorthand for $\land$ e.g. $(\forall x \in { 1, 2, … 10 }) x > 0 \equiv 1 > 0 \land 2 > 0 \land … 10 > 0$</li>
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</ul></li>
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</ul></li>
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<li><p><span class="math">\(\exists\)</span>:</p>
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<li><p>$\exists$:</p>
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<ul>
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<ul>
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<li>Shorthand for <span class="math">\(\lor\)</span> e.g. <span class="math">\((\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5\)</span></li>
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<li>Shorthand for $\lor$ e.g. $(\exists x \in { 1, 2, … 10 }) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor … 10 > 5$</li>
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</ul></li>
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</ul></li>
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<li><p><span class="math">\(\neg \forall \equiv \exists\)</span>, vice versa</p></li>
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<li><p>$\neg \forall \equiv \exists$, vice versa</p></li>
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<li><p>With quantifiers we can write logical statements e.g.</p>
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<li><p>With quantifiers we can write logical statements e.g.</p>
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<ul>
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<ul>
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<li><span class="math">\((\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y\)</span> is a statement and is false</li>
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<li>$(\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y$ is a statement and is false</li>
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<li><span class="math">\((\forall x) (\exists y) x > y\)</span> is true</li>
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<li>$(\forall x) (\exists y) x > y$ is true</li>
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<li>shortcut: <span class="math">\((\exists x, y) \equiv (\exists x) (\exists y)\)</span></li>
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<li>shortcut: $(\exists x, y) \equiv (\exists x) (\exists y)$</li>
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</ul></li>
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</ul></li>
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<li><p>Quantifiers can be expressed in set language, sort of a definition in terms of sets:</p>
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<li><p>Quantifiers can be expressed in set language, sort of a definition in terms of sets:</p>
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<ul>
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<ul>
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<li><span class="math">\((\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}\)</span></li>
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<li>$(\forall x \in \mathbb{X}) (\phi(x)) \equiv { p \in \mathbb{X} : \phi(p) } = \mathbb{X}$</li>
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<li><span class="math">\((\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset\)</span></li>
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<li>$(\exists x \in \mathbb{X}) (\phi(x)) \equiv { q \in \mathbb{X} : \phi(q) } \neq \emptyset$</li>
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<li><span class="math">\((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\)</span></li>
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<li>$(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( { p \in \mathbb{X} : \phi(p) } = \mathbb{X} )$</li>
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</ul></li>
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</ul></li>
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<li><p>Order of quantifiers matters.</p></li>
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<li><p>Order of quantifiers matters.</p></li>
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@ -332,53 +334,53 @@
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<li><p>Cartesian product:</p>
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<li><p>Cartesian product:</p>
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<ul>
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<ul>
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<li><span class="math">\(A \times B = \{ (p, q) : p \in A \land q \in B \}\)</span></li>
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<li>$A \times B = { (p, q) : p \in A \land q \in B }$</li>
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</ul></li>
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</ul></li>
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<li><p>Def: A relation <span class="math">\(R\)</span> on a set <span class="math">\(\mathbb X\)</span> is a subset of <span class="math">\(\mathbb X \times \mathbb X\)</span> (<span class="math">\(R \subseteq \mathbb X \times \mathbb X\)</span>)</p></li>
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|
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<li><p>Def: A relation $R$ on a set $\mathbb X$ is a subset of $\mathbb X \times \mathbb X$ ($R \subseteq \mathbb X \times \mathbb X$)</p></li>
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<li><p>Graph of a function <span class="math">\(f()\)</span>: <span class="math">\(\{ (x, f(x) : x \in Dom(f) \}\)</span></p></li>
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<li><p>Graph of a function $f()$: ${ (x, f(x) : x \in Dom(f) }$</p></li>
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<li><p>Properties of:</p>
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<li><p>Properties of:</p>
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<ul>
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<ul>
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<li>Reflexivity: <span class="math">\((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)</span></li>
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<li>Reflexivity: $(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$</li>
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|
<li>Symmetricity: <span class="math">\([ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]\)</span></li>
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|
<li>Symmetricity: $[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$</li>
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<li>Transitivity: <span class="math">\((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)</span></li>
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<li>Transitivity: $(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$</li>
|
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|
<li>Antisymmetricity: <span class="math">\((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)</span></li>
|
|
|
|
<li>Antisymmetricity: $(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$</li>
|
|
|
|
</ul></li>
|
|
|
|
</ul></li>
|
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<li><p>Equivalence relations:</p>
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|
|
<li><p>Equivalence relations:</p>
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|
|
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<ul>
|
|
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<ul>
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|
|
<li>Def: <span class="math">\(R \subseteq \mathbb X \times \mathbb X\)</span> is said to be an equivalence relation <em>iff</em> <span class="math">\(R\)</span> is reflexive, symmetric and transitive.</li>
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|
<li>Def: $R \subseteq \mathbb X \times \mathbb X$ is said to be an equivalence relation <em>iff</em> $R$ is reflexive, symmetric and transitive.</li>
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|
|
<li>Congruence modulo n: <span class="math">\(p R q \equiv n | p - q\)</span></li>
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|
|
<li>Congruence modulo n: $p R q \equiv n | p - q$</li>
|
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|
|
<li>Def R - and equivalence relation of <span class="math">\(\mathbb X\)</span>: The <em>equivalence class</em> of an element <span class="math">\(x \in \mathbb X\)</span> is the set <span class="math">\([x]_R = \{ y \in \mathbb X : x R y \}\)</span>
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<li>Def R - and equivalence relation of $\mathbb X$: The <em>equivalence class</em> of an element $x \in \mathbb X$ is the set $[x]_R = { y \in \mathbb X : x R y }$
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<ul>
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<ul>
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|
<li>Every <span class="math">\(x \in \mathbb X\)</span> belongs to the equivalence class of some element <span class="math">\(a\)</span>.</li>
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|
<li>Every $x \in \mathbb X$ belongs to the equivalence class of some element $a$.</li>
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|
<li><span class="math">\((\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])\)</span></li>
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<li>$(\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])$</li>
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</ul></li>
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</ul></li>
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</ul></li>
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</ul></li>
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<li><p>Partitions</p>
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<li><p>Partitions</p>
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<ul>
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<ul>
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|
<li><p>A partition is a set containing subsets of some set <span class="math">\(\mathbb X\)</span> such that their collective symmetric difference equals <span class="math">\(\mathbb X\)</span>. A partition of is a set <span class="math">\(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\)</span> such that:</p>
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<li><p>A partition is a set containing subsets of some set $\mathbb X$ such that their collective symmetric difference equals $\mathbb X$. A partition of is a set ${ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X }$ such that:</p>
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<ul>
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<ul>
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<li><span class="math">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)\)</span></li>
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<li>$(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$</li>
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<li><span class="math">\((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\)</span></li>
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<li>$(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$</li>
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</ul></li>
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</ul></li>
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|
<li><p><span class="math">\(\{ A_i \}_{i \in \mathbb I}\)</span> is a partition <em>iff</em> there exists an equivalence relation <span class="math">\(R\)</span> on <span class="math">\(\mathbb X\)</span> such that:</p>
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|
<li><p>${ A<em>i }</em>{i \in \mathbb I}$ is a partition <em>iff</em> there exists an equivalence relation $R$ on $\mathbb X$ such that:</p>
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<ul>
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<ul>
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<li><span class="math">\((\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R\)</span></li>
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<li>$(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$</li>
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<li><span class="math">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j\)</span></li>
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<li>$(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$</li>
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</ul></li>
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</ul></li>
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<li><p>The quotient set: <span class="math">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></p></li>
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<li><p>The quotient set: $\mathbb X / R = { [a] : a \in \mathbb X }$</p></li>
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</ul></li>
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</ul></li>
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</ul>
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</ul>
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@ -388,9 +390,9 @@
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<li><p>Partial orders</p>
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<li><p>Partial orders</p>
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<ul>
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<ul>
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<li><span class="math">\(\mathbb X\)</span> is a set, <span class="math">\(R \subseteq \mathbb X \times \mathbb X\)</span></li>
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<li>$\mathbb X$ is a set, $R \subseteq \mathbb X \times \mathbb X$</li>
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<li><p>Def: <span class="math">\(R\)</span> is a partial order on <span class="math">\(\mathbb X\)</span> iff <span class="math">\(R\)</span> is:</p>
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<li><p>Def: $R$ is a partial order on $\mathbb X$ iff $R$ is:</p>
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<ul>
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<ul>
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<li>Reflexive</li>
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<li>Reflexive</li>
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|
@ -398,24 +400,24 @@
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<li>Transitive</li>
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<li>Transitive</li>
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</ul></li>
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</ul></li>
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<li><p>Def: <span class="math">\(m \in \mathbb X\)</span> is said to be:</p>
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<li><p>Def: $m \in \mathbb X$ is said to be:</p>
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<ul>
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<ul>
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<li>maximal element in <span class="math">\((\mathbb X, \preccurlyeq)\)</span> iff <span class="math">\((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)</span></li>
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|
<li>maximal element in $(\mathbb X, \preccurlyeq)$ iff $(\forall a \in \mathbb X) m \preccurlyeq a \implies m = a$</li>
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|
<li>largest iff <span class="math">\((\forall a \in \mathbb X) (a \preccurlyeq m)\)</span></li>
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<li>largest iff $(\forall a \in \mathbb X) (a \preccurlyeq m)$</li>
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<li>minimal iff <span class="math">\((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)</span></li>
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<li>minimal iff $(\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)$</li>
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|
<li>smallest iff <span class="math">\((\forall a \in \mathbb X) (m \preccurlyeq a)\)</span></li>
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|
<li>smallest iff $(\forall a \in \mathbb X) (m \preccurlyeq a)$</li>
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</ul></li>
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</ul></li>
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<li><p>Def: A partial order <span class="math">\(R\)</span> on <span class="math">\(\mathbb X\)</span> is said to be <em>"total"</em> iff <span class="math">\((\forall x, y \in \mathbb X) (x R y \lor y R x)\)</span></p></li>
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|
<li><p>Def: A partial order $R$ on $\mathbb X$ is said to be <em>“total”</em> iff $(\forall x, y \in \mathbb X) (x R y \lor y R x)$</p></li>
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<li><p>Def: A subset <span class="math">\(B\)</span> of <span class="math">\(\mathbb X\)</span> is called a chain <em>"chain"</em> iff <span class="math">\(B\)</span> is totally ordered by <span class="math">\(R\)</span></p>
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<li><p>Def: A subset $B$ of $\mathbb X$ is called a chain <em>“chain”</em> iff $B$ is totally ordered by $R$</p>
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<ul>
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<ul>
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<li><span class="math">\(C(\mathbb X)\)</span> - the set of all chains in <span class="math">\((\mathbb X, R)\)</span></li>
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<li>$C(\mathbb X)$ - the set of all chains in $(\mathbb X, R)$</li>
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<li>A chain <span class="math">\(D\)</span> in <span class="math">\((\mathbb X, R)\)</span> is called a maximal chain iff <span class="math">\(D\)</span> is a maximal element in <span class="math">\((C(\mathbb X), R)\)</span></li>
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<li>A chain $D$ in $(\mathbb X, R)$ is called a maximal chain iff $D$ is a maximal element in $(C(\mathbb X), R)$</li>
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<li><span class="math">\(K \subseteq \mathbb X\)</span> is called an antichain in <span class="math">\((\mathbb X, R)\)</span> iff <span class="math">\((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)</span></li>
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<li>$K \subseteq \mathbb X$ is called an antichain in $(\mathbb X, R)$ iff $(\forall p, q \in K) (p R q \lor q R p \implies p = q)$</li>
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<li>Def: <span class="math">\(R\)</span> is a partial order on <span class="math">\(\mathbb X\)</span>, <span class="math">\(R\)</span> is called a <em>well</em> order iff <span class="math">\(R\)</span> is a total order on <span class="math">\(X\)</span> and every nonempty subset <span class="math">\(A\)</span> of <span class="math">\(\mathbb X\)</span> has the smallest element</li>
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<li>Def: $R$ is a partial order on $\mathbb X$, $R$ is called a <em>well</em> order iff $R$ is a total order on $X$ and every nonempty subset $A$ of $\mathbb X$ has the smallest element</li>
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</ul></li>
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</ul></li>
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</ul></li>
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</ul></li>
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</ul>
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</ul>
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@ -423,40 +425,40 @@
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<h2 id="induction">Induction</h2>
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<h2 id="induction">Induction</h2>
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<ul>
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<ul>
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<li>If <span class="math">\(\phi\)</span> is a propositional function defined on <span class="math">\(\mathbb N\)</span>, if:
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<li>If $\phi$ is a propositional function defined on $\mathbb N$, if:
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<ul>
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<ul>
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<li><span class="math">\(\phi(1)\)</span></li>
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<li>$\phi(1)$</li>
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<li><span class="math">\((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)</span></li>
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<li>$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$</li>
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<li><span class="math">\((\forall k \geq 1) \phi(k)\)</span></li>
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<li>$(\forall k \geq 1) \phi(k)$</li>
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</ul></li>
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</ul></li>
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</ul>
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</ul>
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<h2 id="functions">Functions</h2>
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<h2 id="functions">Functions</h2>
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<ul>
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<ul>
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<li><span class="math">\(f: \mathbb X \to \mathbb Y\)</span></li>
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<li>$f: \mathbb X \to \mathbb Y$</li>
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<li><p>Def: <span class="math">\(f \subseteq \mathbb X \times \mathbb Y\)</span> is said to be a function if:</p>
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<li><p>Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:</p>
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<ul>
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<ul>
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<li><span class="math">\((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)</span></li>
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<li>$(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$</li>
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<li><span class="math">\((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\)</span></li>
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<li>$(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$</li>
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</ul></li>
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</ul></li>
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<li><p>Types of functions <span class="math">\(f: \mathbb X \to \mathbb Y\)</span>:</p>
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<li><p>Types of functions $f: \mathbb X \to \mathbb Y$:</p>
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<ul>
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<ul>
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<li><span class="math">\(f\)</span> is said to be an injection ( 1 to 1 function) iff <span class="math">\((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)</span></li>
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<li>$f$ is said to be an injection ( 1 to 1 function) iff $(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$</li>
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<li><span class="math">\(f\)</span> is said to be a surjection (onto function) iff <span class="math">\((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)</span></li>
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<li>$f$ is said to be a surjection (onto function) iff $(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$</li>
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<li>If <span class="math">\(f^{-1}\)</span> is a function from <span class="math">\(\mathbb Y \to \mathbb X\)</span> then <span class="math">\(f^{-1}\)</span> is called the inverse function for <span class="math">\(f\)</span>
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<li>If $f^{-1}$ is a function from $\mathbb Y \to \mathbb X$ then $f^{-1}$ is called the inverse function for $f$
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<ul>
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<ul>
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<li>Fact: <span class="math">\(f^{-1}\)</span> is a function iff <span class="math">\(f\)</span> is a <em>bijection</em> (1 to 1 and onto)</li>
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<li>Fact: $f^{-1}$ is a function iff $f$ is a <em>bijection</em> (1 to 1 and onto)</li>
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</ul></li>
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</ul></li>
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</ul></li>
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</ul></li>
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<li><p>For some set <span class="math">\(\mathbb A\)</span> the image of <span class="math">\(\mathbb A\)</span> by <span class="math">\(f\)</span> is <span class="math">\(f(\mathbb A) = \{ f(x) : x \in \mathbb A \}\)</span>. We can also define the inverse of an image even when the function itself isn't invertible: <span class="math">\(f^{-1}(\mathbb A)\)</span></p></li>
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<li><p>For some set $\mathbb A$ the image of $\mathbb A$ by $f$ is $f(\mathbb A) = { f(x) : x \in \mathbb A }$. We can also define the inverse of an image even when the function itself isn’t invertible: $f^{-1}(\mathbb A)$</p></li>
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</ul>
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</ul>
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</div>
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</div>
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@ -488,14 +490,17 @@
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.css"
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integrity="sha384-zB1R0rpPzHqg7Kpt0Aljp8JPLqbXI3bhnPWROx27a9N0Ll6ZP/+DiW/UqRcLbRjq" crossorigin="anonymous">
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<script>
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<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.js"
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MathJax = {
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integrity="sha384-y23I5Q6l+B6vatafAwxRu/0oK/79VlbSz7Q9aiSZUvyWYIYsd+qj+o24G5ZU2zJz"
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tex: {
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crossorigin="anonymous"></script>
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inlineMath: [['$', '$'], ['\\(', '\\)']],
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<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/contrib/auto-render.min.js"
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displayMath: [['$$', '$$'], ['\[', '\]']]
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integrity="sha384-kWPLUVMOks5AQFrykwIup5lo0m3iMkkHrD0uJ4H5cjeGihAutqP0yW0J6dpFiVkI" crossorigin="anonymous"
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}
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onload="renderMathInElement(document.body);"></script>
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};
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</script>
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<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
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</script>
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