Propositional Logic

Proposition

A proposition is a declartive sentence that is either true or false. $$\mbox{p}$$

Negation of p

The negation of p has the opposite truth value of p. $$\neg \mbox{p}$$

Conjunction of p and q

The conjunction of p and q is true when both p and q are true and false otherwise. $$\mbox{p} \wedge \mbox{q}$$

Disjunction of p and q

The disjunction of p and q is false when both p and q are false and true otherwise. $$\mbox{p} \lor \mbox{q}$$

Exclusive or of p and q

The exclusive or of p and q is true when exactly one of p and q are true and false otherwise. $$\mbox{p} \oplus \mbox{q}$$

Conditional statement

The conditional statement if p then q is false when p is true and q is false and true otherwise. $$\mbox{p} \rightarrow \mbox{q}$$

Converse

The converse of p $\rightarrow$ q is the conditional statement

$$\mbox{q} \rightarrow \mbox{p}$$

Contrapositive

The contrapositive of p $\rightarrow$ q is the conditional statement

$$\neg \mbox{q} \rightarrow \neg \mbox{p}$$

Inverse

The inverse of p $\rightarrow$ q is the conditional statement

$$\neg \mbox{p} \rightarrow \neg \mbox{q}$$

Biconditional statement

The biconditional statement p if and only if q is true when p and q have the same truth value and false otherwise. $$\mbox{p} \leftrightarrow \mbox{q}$$

Logical Equivalence

compound propositions p and q are logically equivalent if p if and only if q is a tautology, that is the compound proposition is true no matter the truth values of the propositional variables.

$$\mbox{p} \equiv \mbox{q}$$

Propositional Function

Value of a propositional function P at x. Function defined by it's predicate,P and subject, x where x is the subject of the statement and P refers to a property that the subject has. $$P(x)$$

Propositional Multivariable Function

Value of a propositional function P at the n-tuple $(x_{1},x_{2},\dots,x_{n})$. $$P(x_{1},x_{2},\dots,x_{n})$$

Universal Quantification

Universal quantification of P(x) is true when P(x) is true for every x and false when there is an x for which P(x) is false. $$\forall x P(x)$$

Existential Quanification

Existential quantification of P(x) is true when there exists an element x in the domain such that P(x) and false when P(x) is false for every x. $$\exists x P(x)$$

Quantifaction Equivalences

Statement is true when there is an x for which P(x) is false and false when P(x) is true for every x.

$$\neg \forall x P(x)\equiv\exists x \neg P(x)$$ Statement is true when for every x P(x) is true and false when there is an x for which P(x) is true. $$\neg \exists x P(x)\equiv\forall x \neg P(x)$$

Quantification of Two Variables

P(x,y) is true for every pair x,y and false when there is a pair x,y for which P(x,y) is false.

$$\forall x \forall y P(x,y)$$ $$\forall y \forall x P(x,y)$$ For every x there is a y for which P(x,y) is true. There is an x such that P(x,y) is false for every y. $$\forall x \exists y P(x,y)$$ There is an x for which P(x,y) is true for every y.For every x there is a y for which P(x,y) is false. $$\exists x \forall y P(x,y)$$ There is a pair x,y for which P(x,y) is true. P(x,y) is false for every pair x,y. $$\exists x \exists y P(x,y)$$

$$\exists y \exists x P(x,y)$$