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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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$2=7$ 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://abdulocra.cy/posts/eidma/">Introduction to Discrete Mathematics</a></h1>
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2019-11-20
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#<a href="https://abdulocra.cy/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>$2=7$ statement</li>
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<li>$x=5$ not a statement</li>
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</ul></li>
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||
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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>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: $p \land q$ is only true if both p and q are true $(0001)$</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): $p \implies q$ is false only if p is true and q is false (truth table $(1011)$)
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<ul>
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<li>$\equiv \neg p \lor q$</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 ~, $\neg$ or NOT, negates the one input $(10)$.</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. $\neg[(p \lor q)] \land r$</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: $\phi(p, q, k) \equiv \psi(p, q, k)$, logical value of $\phi$ is equal to logical value of $\psi$.</li>
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<li>Commutativity: $p \land q \equiv q \land p$</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: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$</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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<ul>
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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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<li><p>Double negation law: $\neg(\neg p) \equiv p$</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>): $p \iff p \equiv (p \implies q) \land (q \implies p)$</p></li>
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<li><p>Contraposition law:</p>
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<ul>
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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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<li>$(p \implies q) \equiv (\neg p \lor q)$</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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<li><p>Contradiction law:</p>
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<ul>
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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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<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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<h2 id="sets">Sets</h2>
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<ul>
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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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<li>$2^\mathbb X = { A : A \subseteq \mathbb X}$</li>
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<li>$2^\mathbb X = P(\mathbb X)$</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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<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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<ul>
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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 - $\cap$ - $A \cap B = { x \in \mathbb X : x \in A \land x \in B }$</li>
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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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<li>If $x = { 1 }$ then $x’ = \emptyset$</li>
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</ul></li>
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</ul></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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<ul>
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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: $A \div B = (A \setminus B) \cup (B \setminus A)$</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>$A \cup B = B \cup A , A \cap B = B \cap A$</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>$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ vice versa</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>$(A \cup B)’ = A’ \cap B’$ vice versa</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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<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: ${ x \in \mathbb R : x^2 = -1} = \emptyset$</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>$\phi$ - prepositional function: yields only true or false value</li>
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<li>$\forall$ means “for all” and $\exists$ means “there exists”</li>
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<li><p>$\forall$:</p>
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<ul>
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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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<li><p>$\exists$:</p>
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<ul>
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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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<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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<ul>
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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>$(\forall x) (\exists y) x > y$ is true</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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<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>$(\forall x \in \mathbb{X}) (\phi(x)) \equiv { p \in \mathbb{X} : \phi(p) } = \mathbb{X}$</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>$(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( { p \in \mathbb{X} : \phi(p) } = \mathbb{X} )$</li>
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</ul></li>
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<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>$A \times B = { (p, q) : p \in A \land q \in B }$</li>
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</ul></li>
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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 $f()$: ${ (x, f(x) : x \in Dom(f) }$</p></li>
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<li><p>Properties of:</p>
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<ul>
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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: $[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$</li>
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<li>Transitivity: $(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$</li>
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<li>Antisymmetricity: $(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$</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: $R \subseteq \mathbb X \times \mathbb X$ 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: $p R q \equiv n | p - q$</li>
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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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<li>Every $x \in \mathbb X$ belongs to the equivalence class of some element $a$.</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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<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 $\mathbb X$ 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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<li>$(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$</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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<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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<li>$(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$</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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<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>
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<h2 id="posets">Posets</h2>
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<ul>
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<li><p>Partial orders</p>
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<ul>
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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: $R$ is a partial order on $\mathbb X$ iff $R$ is:</p>
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<ul>
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<li>Reflexive</li>
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<li>Antisymmetric</li>
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<li>Transitive</li>
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</ul></li>
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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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<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 $(\forall a \in \mathbb X) (a \preccurlyeq m)$</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 $(\forall a \in \mathbb X) (m \preccurlyeq a)$</li>
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</ul></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 $B$ 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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<li>$C(\mathbb X)$ - the set of all chains in $(\mathbb X, R)$</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>$K \subseteq \mathbb X$ 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: $R$ 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>
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<h2 id="induction">Induction</h2>
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<ul>
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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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<li>$\phi(1)$</li>
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<li>$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$</li>
|
||
<li>$(\forall k \geq 1) \phi(k)$</li>
|
||
</ul></li>
|
||
</ul>
|
||
|
||
<h2 id="functions">Functions</h2>
|
||
|
||
<ul>
|
||
<li>$f: \mathbb X \to \mathbb Y$</li>
|
||
|
||
<li><p>Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:</p>
|
||
|
||
<ul>
|
||
<li>$(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$</li>
|
||
<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>
|
||
</ul></li>
|
||
|
||
<li><p>Types of functions $f: \mathbb X \to \mathbb Y$:</p>
|
||
|
||
<ul>
|
||
<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>
|
||
<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>
|
||
<li>If $f^{-1}$ is a function from $\mathbb Y \to \mathbb X$ then $f^{-1}$ is called the inverse function for $f$
|
||
|
||
<ul>
|
||
<li>Fact: $f^{-1}$ is a function iff $f$ is a <em>bijection</em> (1 to 1 and onto)</li>
|
||
</ul></li>
|
||
</ul></li>
|
||
|
||
<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>
|
||
</ul>
|
||
|
||
</div>
|
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|
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