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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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\(2=7\) 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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<h1 class="post-title">
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<a href="https://022385.xyz/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
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2019-11-04
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<span class="post-author">::
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Abdulkadir
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<span class="post-tags">
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#<a href="https://022385.xyz/tags/university-notes/">university-notes</a>
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<div class="post-content">
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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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</ul>
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<h2 id="propositional-calculus">Propositional calculus</h2>
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<ul>
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<li>Comes from the linguistic concept that things can be either true or false.</li>
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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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<li><span class="math">\(2=7\)</span> statement</li>
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<li><span class="math">\(x=5\)</span> not a statement</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>Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).</p></li>
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<li><p>When doing logic, we use propositional variables (e.g. p, q, r).</p>
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<ul>
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<li>Can be either <strong>true</strong> or <strong>false</strong>.</li>
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</ul></li>
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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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<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>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>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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<ul>
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<li><span class="math">\(\equiv \neg p \lor q\)</span></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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<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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</ul></li>
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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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<li>e.g. <span class="math">\(\neg[(p \lor q)] \land r\)</span></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>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>Commutativity: <span class="math">\(p \land q \equiv q \land p\)</span></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>Distributivity: <span class="math">\(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\)</span></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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<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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</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>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>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>Contraposition law:</p>
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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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<ul>
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<li><span class="math">\((p \implies q) \equiv (\neg p \lor q)\)</span></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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</ul></li>
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</ul></li>
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<li><p>Contradiction law:</p>
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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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</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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</ul>
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<h2 id="sets">Sets</h2>
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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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<ul>
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<li><span class="math">\(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\)</span></li>
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<li><span class="math">\(2^\mathbb X = P(\mathbb X)\)</span></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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</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>Operations on sets:</p>
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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>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>Complement - <span class="math">\(A'\)</span> - <span class="math">\(A' = \{ x \in \mathbb X : \neg (x \in A) \}\)</span>
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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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</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>Difference of sets:</p>
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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>Symmetric difference: <span class="math">\(A \div B = (A \setminus B) \cup (B \setminus A)\)</span></li>
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</ul></li>
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<li><p>Laws of set algebra:</p>
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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><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><span class="math">\((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)</span> 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><span class="math">\((A \cup B)' = A' \cap B'\)</span> 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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</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>Quip: <span class="math">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</span></p></li>
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</ul>
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<h2 id="quantifiers">Quantifiers</h2>
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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><span class="math">\(\forall\)</span> means "for all" and <span class="math">\(\exists\)</span> means "there exists"</li>
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<li><p><span class="math">\(\forall\)</span>:</p>
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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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</ul></li>
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<li><p><span class="math">\(\exists\)</span>:</p>
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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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</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>With quantifiers we can write logical statements e.g.</p>
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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><span class="math">\((\forall x) (\exists y) x > y\)</span> 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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</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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<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><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><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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</ul></li>
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<li><p>Order of quantifiers matters.</p></li>
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</ul>
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<h2 id="relations">Relations</h2>
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<ul>
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<li><p>Cartesian product:</p>
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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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</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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<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>Properties of:</p>
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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>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>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>Antisymmetricity: <span class="math">\((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)</span></li>
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</ul></li>
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<li><p>Equivalence relations:</p>
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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>Congruence modulo n: <span class="math">\(p R q \equiv n | p - q\)</span></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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<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><span class="math">\((\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])\)</span></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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<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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<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><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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</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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<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><span class="math">\((\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j\)</span></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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