<li>Conjunction: <spanclass="math">\(p \land q\)</span> is only true if both p and q are true <spanclass="math">\((0001)\)</span></li>
<li>Disjunction: <spanclass="math">\(p \lor q\)</span> is only false if both p and q are false <spanclass="math">\((0111)\)</span></li>
<li>Implication (material conditional): <spanclass="math">\(p \implies q\)</span> is false only if p is true and q is false (truth table <spanclass="math">\((1011)\)</span>)
<ul>
<li><spanclass="math">\(\equiv \neg p \lor q\)</span></li>
<li><spanclass="math">\((p \land)\)</span> is not a formula, as <spanclass="math">\(\land\)</span> requires 2 variables.</li>
<li>Logical equivalence: <spanclass="math">\(\phi(p, q, k) \equiv \psi(p, q, k)\)</span>, logical value of <spanclass="math">\(\phi\)</span> is equal to logical value of <spanclass="math">\(\psi\)</span>.</li>
<li><p><spanclass="math">\(A \subset B \equiv\)</span> every element of A is an element of B <spanclass="math">\(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)</span></p></li>
<li><p>Note: <spanclass="math">\(\{ \emptyset \} \neq \emptyset\)</span>, one is a set with one element, one is the empty set, no elements (<spanclass="math">\(\{ \}\)</span>)</p></li>
<li><p>Quip: <spanclass="math">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</span></p></li>
<li><p>Def: A relation <spanclass="math">\(R\)</span> on a set <spanclass="math">\(\mathbb X\)</span> is a subset of <spanclass="math">\(\mathbb X \times \mathbb X\)</span> (<spanclass="math">\(R \subseteq \mathbb X \times \mathbb X\)</span>)</p></li>
<li><p>Graph of a function <spanclass="math">\(f()\)</span>: <spanclass="math">\(\{ (x, f(x) : x \in Dom(f) \}\)</span></p></li>
<li>Reflexivity: <spanclass="math">\((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)</span></li>
<li>Symmetricity: <spanclass="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>
<li>Transitivity: <spanclass="math">\((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)</span></li>
<li>Antisymmetricity: <spanclass="math">\((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)</span></li>
<li>Def: <spanclass="math">\(R \subseteq \mathbb X \times \mathbb X\)</span> is said to be an equivalence relation <em>iff</em><spanclass="math">\(R\)</span> is reflexive, symmetric and transitive.</li>
<li>Congruence modulo n: <spanclass="math">\(p R q \equiv n | p - q\)</span></li>
<li>Def R - and equivalence relation of <spanclass="math">\(\mathbb X\)</span>: The <em>equivalence class</em> of an element <spanclass="math">\(x \in \mathbb X\)</span> is the set <spanclass="math">\([x]_R = \{ y \in \mathbb X : x R y \}\)</span>
<ul>
<li>Every <spanclass="math">\(x \in \mathbb X\)</span> belongs to the equivalence class of some element <spanclass="math">\(a\)</span>.</li>
<li><p>A partition is a set containing subsets of some set <spanclass="math">\(\mathbb X\)</span> such that their collective symmetric difference equals <spanclass="math">\(\mathbb X\)</span>. A partition of is a set <spanclass="math">\(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\)</span> such that:</p>
<li><p><spanclass="math">\(\{ A_i \}_{i \in \mathbb I}\)</span> is a partition <em>iff</em> there exists an equivalence relation <spanclass="math">\(R\)</span> on <spanclass="math">\(\mathbb X\)</span> such that:</p>
<li><spanclass="math">\(\mathbb X\)</span> is a set, <spanclass="math">\(R \subseteq \mathbb X \times \mathbb X\)</span></li>
<li><p>Def: <spanclass="math">\(R\)</span> is a partial order on <spanclass="math">\(\mathbb X\)</span> iff <spanclass="math">\(R\)</span> is:</p>
<ul>
<li>Reflexive</li>
<li>Antisymmetric</li>
<li>Transitive</li>
</ul></li>
<li><p>Def: <spanclass="math">\(m \in \mathbb X\)</span> is said to be:</p>
<ul>
<li>maximal element in <spanclass="math">\((\mathbb X, \preccurlyeq)\)</span> iff <spanclass="math">\((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)</span></li>
<li>largest iff <spanclass="math">\((\forall a \in \mathbb X) (a \preccurlyeq m)\)</span></li>
<li>minimal iff <spanclass="math">\((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)</span></li>
<li>smallest iff <spanclass="math">\((\forall a \in \mathbb X) (m \preccurlyeq a)\)</span></li>
</ul></li>
<li><p>Def: A partial order <spanclass="math">\(R\)</span> on <spanclass="math">\(\mathbb X\)</span> is said to be <em>"total"</em> iff <spanclass="math">\((\forall x, y \in \mathbb X) (x R y \lor y R x)\)</span></p></li>
<li><p>Def: A subset <spanclass="math">\(B\)</span> of <spanclass="math">\(\mathbb X\)</span> is called a chain <em>"chain"</em> iff <spanclass="math">\(B\)</span> is totally ordered by <spanclass="math">\(R\)</span></p>
<ul>
<li><spanclass="math">\(C(\mathbb X)\)</span> - the set of all chains in <spanclass="math">\((\mathbb X, R)\)</span></li>
<li>A chain <spanclass="math">\(D\)</span> in <spanclass="math">\((\mathbb X, R)\)</span> is called a maximal chain iff <spanclass="math">\(D\)</span> is a maximal element in <spanclass="math">\((C(\mathbb X), R)\)</span></li>
<li><spanclass="math">\(K \subseteq \mathbb X\)</span> is called an antichain in <spanclass="math">\((\mathbb X, R)\)</span> iff <spanclass="math">\((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)</span></li>
<li>Def: <spanclass="math">\(R\)</span> is a partial order on <spanclass="math">\(\mathbb X\)</span>, <spanclass="math">\(R\)</span> is called a <em>well</em> order iff <spanclass="math">\(R\)</span> is a total order on <spanclass="math">\(X\)</span> and every nonempty subset <spanclass="math">\(A\)</span> of <spanclass="math">\(\mathbb X\)</span> has the smallest element</li>
</ul></li>
</ul></li>
</ul>
<h2id="induction">Induction</h2>
<ul>
<li>If <spanclass="math">\(\phi\)</span> is a propositional function defined on <spanclass="math">\(\mathbb N\)</span>, if:
<li><spanclass="math">\(f: \mathbb X \to \mathbb Y\)</span></li>
<li><p>Def: <spanclass="math">\(f \subseteq \mathbb X \times \mathbb Y\)</span> is said to be a function if:</p>
<ul>
<li><spanclass="math">\((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)</span></li>
<li><spanclass="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>
</ul></li>
<li><p>Types of functions <spanclass="math">\(f: \mathbb X \to \mathbb Y\)</span>:</p>
<ul>
<li><spanclass="math">\(f\)</span> is said to be an injection ( 1 to 1 function) iff <spanclass="math">\((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)</span></li>
<li><spanclass="math">\(f\)</span> is said to be a surjection (onto function) iff <spanclass="math">\((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)</span></li>
<li>If <spanclass="math">\(f^{-1}\)</span> is a function from <spanclass="math">\(\mathbb Y \to \mathbb X\)</span> then <spanclass="math">\(f^{-1}\)</span> is called the inverse function for <spanclass="math">\(f\)</span>
<ul>
<li>Fact: <spanclass="math">\(f^{-1}\)</span> is a function iff <spanclass="math">\(f\)</span> is a <em>bijection</em> (1 to 1 and onto)</li>
</ul></li>
</ul></li>
<li><p>For some set <spanclass="math">\(\mathbb A\)</span> the image of <spanclass="math">\(\mathbb A\)</span> by <spanclass="math">\(f\)</span> is <spanclass="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: <spanclass="math">\(f^{-1}(\mathbb A)\)</span></p></li>
