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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>We should avoid variables when forming statements, as they may change the logical value.
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<ul>
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<li><span class="math inline">2=7</span> statement</li>
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<li><span class="math inline">x=5</span> not a statement</li>
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</ul></li>
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<li>In logic we do not use the equals sign, we use the equivalence sign <span class="math inline">\equiv</span>.</li>
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<li>Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).</li>
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<li>When doing logic, we use propositional variables (e.g. p, q, r).
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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>The operations done on propositional variables are called propositional connectives.
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<ul>
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<li>Conjunction: <span class="math inline">p \land q</span> is only true if both p and q are true <span class="math inline">(0001)</span></li>
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<li>Disjunction: <span class="math inline">p \lor q</span> is only false if both p and q are false <span class="math inline">(0111)</span></li>
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<li>Implication (material conditional): <span class="math inline">p \implies q</span> is false only if p is true and q is false (truth table <span class="math inline">(1011)</span>)
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<ul>
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<li><span class="math inline">\equiv \neg p \lor q</span></li>
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</ul></li>
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</ul></li>
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<li>Not necessarily connectives but unary operations:
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<ul>
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<li>Negation: Denoted by ~, <span class="math inline">\neg</span> or NOT, negates the one input <span class="math inline">(10)</span>.</li>
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</ul></li>
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<li>A (propositional) formula is a “properly constructed” logical expression.
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<ul>
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<li>e.g. <span class="math inline">\neg[(p \lor q)] \land r</span></li>
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<li><span class="math inline">(p \land)</span> is not a formula, as <span class="math inline">\land</span> requires 2 variables.</li>
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<li>Logical equivalence: <span class="math inline">\phi(p, q, k) \equiv \psi(p, q, k)</span>, logical value of <span class="math inline">\phi</span> is equal to logical value of <span class="math inline">\psi</span>.</li>
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<li>Commutativity: <span class="math inline">p \land q \equiv q \land p</span></li>
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<li>Associativity: <span class="math inline">(p \land q) \land r \equiv p \land (q \land r)</span></li>
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<li>Distributivity: <span class="math inline">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 inline">\neg(B \lor C)</span> can be written as <span class="math inline">\neg B \land \neg C</span></li>
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</ul></li>
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</ul></li>
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<li>Double negation law: <span class="math inline">\neg(\neg p) \equiv p</span></li>
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<li><p>De Morgan’s laws: <span class="math inline">\neg(p \land q) \equiv \neg p \lor \neg q</span> and <span class="math inline">\neg(p \lor q) \equiv \neg p \land \neg q</span>.</p></li>
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<li>If and only if (<em>iff</em>): <span class="math inline">p \iff p \equiv (p \implies q) \land (q \implies p)</span></li>
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<li>Contraposition law:
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<ul>
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<li><span class="math inline">(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 inline">(p \implies q) \equiv (\neg p \lor q)</span></li>
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<li><span class="math inline">(\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>Contradiction law:
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<ul>
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<li><span class="math inline">p \lor \neg p \equiv 1</span> and <span class="math inline">p \land \neg p \equiv 0</span></li>
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</ul></li>
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<li><p>Tautology: <span class="math inline">\phi (p, q, ... r)</span> is a tautology <em>iff</em> <span class="math inline">\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>We will consider subsets of universal set <span class="math inline">\mathbb X</span>
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<ul>
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<li><span class="math inline">2^\mathbb X = \{ A : A \subseteq \mathbb X\}</span></li>
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<li><span class="math inline">2^\mathbb X = P(\mathbb X)</span></li>
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<li>All 2 object subsets of <span class="math inline">\mathbb X</span>: <span class="math inline">P_2(\mathbb X)</span></li>
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</ul></li>
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<li><span class="math inline">A \subset B \equiv</span> every element of A is an element of B <span class="math inline">\equiv \{x \in \mathbb X : x \in A \implies x \in B\}</span></li>
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<li>Operations on sets:
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<ul>
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<li>Union - <span class="math inline">\cup</span> - <span class="math inline">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 inline">\cap</span> - <span class="math inline">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 inline">A'</span> - <span class="math inline">A' = \{ x \in \mathbb X : \neg (x \in A) \}</span>
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<ul>
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<li>If <span class="math inline">x = \{ 1 \}</span> then <span class="math inline">x' = \emptyset</span></li>
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</ul></li>
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</ul></li>
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<li>Equality of sets: <span class="math inline">A = B</span> iff <span class="math inline">x \in \mathbb X : (x \in A \iff x \in B)</span></li>
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<li>Difference of sets:
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<ul>
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<li><span class="math inline">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 inline">A \div B = (A \setminus B) \cup (B \setminus A)</span></li>
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</ul></li>
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<li>Laws of set algebra:
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<ul>
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<li><span class="math inline">A \cup B = B \cup A , A \cap B = B \cap A</span></li>
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<li><span class="math inline">(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 inline">(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)</span> vice versa</li>
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<li><span class="math inline">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 inline">(A \cup B)' = A' \cap B'</span> vice versa</li>
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<li><span class="math inline">A \cup A' = \mathbb X, A \cap A' = \emptyset</span></li>
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</ul></li>
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<li>Note: <span class="math inline">\{ \emptyset \} \neq \emptyset</span>, one is a set with one element, one is the empty set, no elements (<span class="math inline">\{ \}</span>)</li>
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<li>Quip: <span class="math inline">\{ x \in \mathbb R : x^2 = -1\} = \emptyset</span></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 inline">\phi</span> - prepositional function: yields only true or false value</li>
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<li><span class="math inline">\forall</span> means “for all” and <span class="math inline">\exists</span> means “there exists”</li>
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<li><span class="math inline">\forall</span>:
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<ul>
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<li>Shorthand for <span class="math inline">\land</span> e.g. <span class="math inline">(\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><span class="math inline">\exists</span>:
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<ul>
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<li>Shorthand for <span class="math inline">\lor</span> e.g. <span class="math inline">(\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><span class="math inline">\neg \forall \equiv \exists</span>, vice versa</li>
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<li>With quantifiers we can write logical statements e.g.
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<ul>
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<li><span class="math inline">(\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 inline">(\forall x) (\exists y) x > y</span> is true</li>
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<li>shortcut: <span class="math inline">(\exists x, y) \equiv (\exists x) (\exists y)</span></li>
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</ul></li>
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<li>Quantifiers can be expressed in set language, sort of a definition in terms of sets:
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<ul>
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<li><span class="math inline">(\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 inline">(\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 inline">(\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>Order of quantifiers matters.</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>Cartesian product:
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<ul>
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<li><span class="math inline">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>Def: A relation <span class="math inline">R</span> on a set <span class="math inline">\mathbb X</span> is a subset of <span class="math inline">\mathbb X \times \mathbb X</span> (<span class="math inline">R \subseteq \mathbb X \times \mathbb X</span>)</li>
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<li>Graph of a function <span class="math inline">f()</span>: <span class="math inline">\{ (x, f(x) : x \in Dom(f) \}</span></li>
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<li>Properties of:
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<ul>
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<li>Reflexivity: <span class="math inline">(\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 inline">[ (\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 inline">(\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 inline">(\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>Equivalence relations:
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<ul>
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<li>Def: <span class="math inline">R \subseteq \mathbb X \times \mathbb X</span> is said to be an equivalence relation <em>iff</em> <span class="math inline">R</span> is reflexive, symmetric and transitive.</li>
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<li>Congruence modulo n: <span class="math inline">p R q \equiv n | p - q</span></li>
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<li>Def R - and equivalence relation of <span class="math inline">\mathbb X</span>: The <em>equivalence class</em> of an element <span class="math inline">x \in \mathbb X</span> is the set <span class="math inline">[x]_R = \{ y \in \mathbb X : x R y \}</span>
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<ul>
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<li>Every <span class="math inline">x \in \mathbb X</span> belongs to the equivalence class of some element <span class="math inline">a</span>.</li>
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<li><span class="math inline">(\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>Partitions
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<ul>
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<li>A partition is a set containing subsets of some set <span class="math inline">\mathbb X</span> such that their collective symmetric difference equals <span class="math inline">\mathbb X</span>. A partition of is a set <span class="math inline">\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}</span> such that:
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<ul>
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<li><span class="math inline">(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)</span></li>
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<li><span class="math inline">(\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><span class="math inline">\{ A_i \}_{i \in \mathbb I}</span> is a partition <em>iff</em> there exists an equivalence relation <span class="math inline">R</span> on <span class="math inline">\mathbb X</span> such that:
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<ul>
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<li><span class="math inline">(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R</span></li>
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<li><span class="math inline">(\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>The quotient set: <span class="math inline">\mathbb X / R = \{ [a] : a \in \mathbb X \}</span></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>Partial orders
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<ul>
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<li><span class="math inline">\mathbb X</span> is a set, <span class="math inline">R \subseteq \mathbb X \times \mathbb X</span></li>
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<li>Def: <span class="math inline">R</span> is a partial order on <span class="math inline">\mathbb X</span> iff <span class="math inline">R</span> is:
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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>Def: <span class="math inline">m \in \mathbb X</span> is said to be:
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<ul>
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<li>maximal element in <span class="math inline">(\mathbb X, \preccurlyeq)</span> iff <span class="math inline">(\forall a \in \mathbb X) m \preccurlyeq a \implies m = a</span></li>
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<li>largest iff <span class="math inline">(\forall a \in \mathbb X) (a \preccurlyeq m)</span></li>
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<li>minimal iff <span class="math inline">(\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)</span></li>
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<li>smallest iff <span class="math inline">(\forall a \in \mathbb X) (m \preccurlyeq a)</span></li>
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</ul></li>
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<li>Def: A partial order <span class="math inline">R</span> on <span class="math inline">\mathbb X</span> is said to be <em>“total”</em> iff <span class="math inline">(\forall x, y \in \mathbb X) (x R y \lor y R x)</span></li>
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||
<li>Def: A subset <span class="math inline">B</span> of <span class="math inline">\mathbb X</span> is called a chain <em>“chain”</em> iff <span class="math inline">B</span> is totally ordered by <span class="math inline">R</span>
|
||
<ul>
|
||
<li><span class="math inline">C(\mathbb X)</span> - the set of all chains in <span class="math inline">(\mathbb X, R)</span></li>
|
||
<li>A chain <span class="math inline">D</span> in <span class="math inline">(\mathbb X, R)</span> is called a maximal chain iff <span class="math inline">D</span> is a maximal element in <span class="math inline">(C(\mathbb X), R)</span></li>
|
||
<li><span class="math inline">K \subseteq \mathbb X</span> is called an antichain in <span class="math inline">(\mathbb X, R)</span> iff <span class="math inline">(\forall p, q \in K) (p R q \lor q R p \implies p = q)</span></li>
|
||
<li>Def: <span class="math inline">R</span> is a partial order on <span class="math inline">\mathbb X</span>, <span class="math inline">R</span> is called a <em>well</em> order iff <span class="math inline">R</span> is a total order on <span class="math inline">X</span> and every nonempty subset <span class="math inline">A</span> of <span class="math inline">\mathbb X</span> has the smallest element</li>
|
||
</ul></li>
|
||
</ul></li>
|
||
</ul>
|
||
<h2 id="induction">Induction</h2>
|
||
<ul>
|
||
<li>If <span class="math inline">\phi</span> is a propositional function defined on <span class="math inline">\mathbb N</span>, if:
|
||
<ul>
|
||
<li><span class="math inline">\phi(1)</span></li>
|
||
<li><span class="math inline">(\forall n \geq 1) \phi(n) \implies \phi(n+1)</span></li>
|
||
<li><span class="math inline">(\forall k \geq 1) \phi(k)</span></li>
|
||
</ul></li>
|
||
</ul>
|
||
<h2 id="functions">Functions</h2>
|
||
<ul>
|
||
<li><span class="math inline">f: \mathbb X \to \mathbb Y</span></li>
|
||
<li>Def: <span class="math inline">f \subseteq \mathbb X \times \mathbb Y</span> is said to be a function if:
|
||
<ul>
|
||
<li><span class="math inline">(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))</span></li>
|
||
<li><span class="math inline">(\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>
|
||
</ul></li>
|
||
<li>Types of functions <span class="math inline">f: \mathbb X \to \mathbb Y</span>:
|
||
<ul>
|
||
<li><span class="math inline">f</span> is said to be an injection ( 1 to 1 function) iff <span class="math inline">(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)</span></li>
|
||
<li><span class="math inline">f</span> is said to be a surjection (onto function) iff <span class="math inline">(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y</span></li>
|
||
<li>If <span class="math inline">f^{-1}</span> is a function from <span class="math inline">\mathbb Y \to \mathbb X</span> then <span class="math inline">f^{-1}</span> is called the inverse function for <span class="math inline">f</span>
|
||
<ul>
|
||
<li>Fact: <span class="math inline">f^{-1}</span> is a function iff <span class="math inline">f</span> is a <em>bijection</em> (1 to 1 and onto)</li>
|
||
</ul></li>
|
||
</ul></li>
|
||
<li>For some set <span class="math inline">\mathbb A</span> the image of <span class="math inline">\mathbb A</span> by <span class="math inline">f</span> is <span class="math inline">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 inline">f^{-1}(\mathbb A)</span></li>
|
||
</ul>
|
||
<h2 id="combinatorics">Combinatorics</h2>
|
||
<ul>
|
||
<li><span class="math inline">|\mathbb A|</span> size (number of elements) of <span class="math inline">\mathbb A</span></li>
|
||
<li>Rule of addition:
|
||
<ul>
|
||
<li>If <span class="math inline">\mathbb A, \mathbb B \subseteq \mathbb X</span> and <span class="math inline">|\mathbb A|, |\mathbb B| \in \mathbb N</span> and <span class="math inline">\mathbb A \cap \mathbb B = \emptyset</span> then <span class="math inline">|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|</span></li>
|
||
<li>Can be generalized as: <span class="math display">
|
||
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\
|
||
|\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\
|
||
(\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset)
|
||
</span></li>
|
||
</ul></li>
|
||
<li>Rule of multiplication:
|
||
<ul>
|
||
<li><span class="math inline">\mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|</span></li>
|
||
<li>Can be generalized as: <span class="math display">
|
||
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\
|
||
|\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}|
|
||
</span></li>
|
||
</ul></li>
|
||
</ul>
|
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|
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