<metaname="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. \(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\). Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc."/>
<metaname="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. \(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\). Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc."/>
<metaproperty="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. \(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\). Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc."/>
<li>Conjunction: <spanclass="math inline">\(p \land q\)</span> is only true if both p and q are true <spanclass="math inline">\((0001)\)</span></li>
<li>Disjunction: <spanclass="math inline">\(p \lor q\)</span> is only false if both p and q are false <spanclass="math inline">\((0111)\)</span></li>
<li>Implication (material conditional): <spanclass="math inline">\(p \implies q\)</span> is false only if p is true and q is false (truth table <spanclass="math inline">\((1011)\)</span>)
<li><spanclass="math inline">\((p \land)\)</span> is not a formula, as <spanclass="math inline">\(\land\)</span> requires 2 variables.</li>
<li>Logical equivalence: <spanclass="math inline">\(\phi(p, q, k) \equiv \psi(p, q, k)\)</span>, logical value of <spanclass="math inline">\(\phi\)</span> is equal to logical value of <spanclass="math inline">\(\psi\)</span>.</li>
<li><p>Tautology: <spanclass="math inline">\(\phi (p, q, ... r)\)</span> is a tautology <em>iff</em><spanclass="math inline">\(\phi \equiv 1\)</span></p></li>
<li><spanclass="math inline">\(A \subset B \equiv\)</span> every element of A is an element of B <spanclass="math inline">\(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)</span></li>
<li>Union - <spanclass="math inline">\(\cup\)</span> - <spanclass="math inline">\(A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}\)</span></li>
<li>Intersection - <spanclass="math inline">\(\cap\)</span> - <spanclass="math inline">\(A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}\)</span></li>
<li>Complement - <spanclass="math inline">\(A'\)</span> - <spanclass="math inline">\(A' = \{ x \in \mathbb X : \neg (x \in A) \}\)</span>
<li>Note: <spanclass="math inline">\(\{ \emptyset \} \neq \emptyset\)</span>, one is a set with one element, one is the empty set, no elements (<spanclass="math inline">\(\{ \}\)</span>)</li>
<li>Quip: <spanclass="math inline">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</span></li>
<li>Def: A relation <spanclass="math inline">\(R\)</span> on a set <spanclass="math inline">\(\mathbb X\)</span> is a subset of <spanclass="math inline">\(\mathbb X \times \mathbb X\)</span> (<spanclass="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span>)</li>
<li>Graph of a function <spanclass="math inline">\(f()\)</span>: <spanclass="math inline">\(\{ (x, f(x) : x \in Dom(f) \}\)</span></li>
<li>Reflexivity: <spanclass="math inline">\((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)</span></li>
<li>Symmetricity: <spanclass="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>
<li>Transitivity: <spanclass="math inline">\((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)</span></li>
<li>Antisymmetricity: <spanclass="math inline">\((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)</span></li>
<li>Def: <spanclass="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span> is said to be an equivalence relation <em>iff</em><spanclass="math inline">\(R\)</span> is reflexive, symmetric and transitive.</li>
<li>Congruence modulo n: <spanclass="math inline">\(p R q \equiv n | p - q\)</span></li>
<li>Def R - and equivalence relation of <spanclass="math inline">\(\mathbb X\)</span>: The <em>equivalence class</em> of an element <spanclass="math inline">\(x \in \mathbb X\)</span> is the set <spanclass="math inline">\([x]_R = \{ y \in \mathbb X : x R y \}\)</span>
<li>Every <spanclass="math inline">\(x \in \mathbb X\)</span> belongs to the equivalence class of some element <spanclass="math inline">\(a\)</span>.</li>
<li>A partition is a set containing subsets of some set <spanclass="math inline">\(\mathbb X\)</span> such that their collective symmetric difference equals <spanclass="math inline">\(\mathbb X\)</span>. A partition of is a set <spanclass="math inline">\(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\)</span> such that:
<li><spanclass="math inline">\(\{ A_i \}_{i \in \mathbb I}\)</span> is a partition <em>iff</em> there exists an equivalence relation <spanclass="math inline">\(R\)</span> on <spanclass="math inline">\(\mathbb X\)</span> such that:
<li><spanclass="math inline">\(\mathbb X\)</span> is a set, <spanclass="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span></li>
<li>Def: <spanclass="math inline">\(R\)</span> is a partial order on <spanclass="math inline">\(\mathbb X\)</span> iff <spanclass="math inline">\(R\)</span> is:
<li>maximal element in <spanclass="math inline">\((\mathbb X, \preccurlyeq)\)</span> iff <spanclass="math inline">\((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)</span></li>
<li>largest iff <spanclass="math inline">\((\forall a \in \mathbb X) (a \preccurlyeq m)\)</span></li>
<li>minimal iff <spanclass="math inline">\((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)</span></li>
<li>Def: A partial order <spanclass="math inline">\(R\)</span> on <spanclass="math inline">\(\mathbb X\)</span> is said to be <em>“total”</em> iff <spanclass="math inline">\((\forall x, y \in \mathbb X) (x R y \lor y R x)\)</span></li>
<li>Def: A subset <spanclass="math inline">\(B\)</span> of <spanclass="math inline">\(\mathbb X\)</span> is called a chain <em>“chain”</em> iff <spanclass="math inline">\(B\)</span> is totally ordered by <spanclass="math inline">\(R\)</span>
<li><spanclass="math inline">\(C(\mathbb X)\)</span> - the set of all chains in <spanclass="math inline">\((\mathbb X, R)\)</span></li>
<li>A chain <spanclass="math inline">\(D\)</span> in <spanclass="math inline">\((\mathbb X, R)\)</span> is called a maximal chain iff <spanclass="math inline">\(D\)</span> is a maximal element in <spanclass="math inline">\((C(\mathbb X), R)\)</span></li>
<li><spanclass="math inline">\(K \subseteq \mathbb X\)</span> is called an antichain in <spanclass="math inline">\((\mathbb X, R)\)</span> iff <spanclass="math inline">\((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)</span></li>
<li>Def: <spanclass="math inline">\(R\)</span> is a partial order on <spanclass="math inline">\(\mathbb X\)</span>, <spanclass="math inline">\(R\)</span> is called a <em>well</em> order iff <spanclass="math inline">\(R\)</span> is a total order on <spanclass="math inline">\(X\)</span> and every nonempty subset <spanclass="math inline">\(A\)</span> of <spanclass="math inline">\(\mathbb X\)</span> has the smallest element</li>
<li><spanclass="math inline">\(f\)</span> is said to be an injection ( 1 to 1 function) iff <spanclass="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><spanclass="math inline">\(f\)</span> is said to be a surjection (onto function) iff <spanclass="math inline">\((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)</span></li>
<li>If <spanclass="math inline">\(f^{-1}\)</span> is a function from <spanclass="math inline">\(\mathbb Y \to \mathbb X\)</span> then <spanclass="math inline">\(f^{-1}\)</span> is called the inverse function for <spanclass="math inline">\(f\)</span>
<li>Fact: <spanclass="math inline">\(f^{-1}\)</span> is a function iff <spanclass="math inline">\(f\)</span> is a <em>bijection</em> (1 to 1 and onto)</li>
<li>For some set <spanclass="math inline">\(\mathbb A\)</span> the image of <spanclass="math inline">\(\mathbb A\)</span> by <spanclass="math inline">\(f\)</span> is <spanclass="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: <spanclass="math inline">\(f^{-1}(\mathbb A)\)</span></li>
<li>If <spanclass="math inline">\(\mathbb A, \mathbb B \subseteq \mathbb X\)</span> and <spanclass="math inline">\(|\mathbb A|, |\mathbb B| \in \mathbb N\)</span> and <spanclass="math inline">\(\mathbb A \cap \mathbb B = \emptyset\)</span> then <spanclass="math inline">\(|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|\)</span></li>
<li>Can be generalized as: <spanclass="math display">\[
