Discrete Definitions

Running Definitions

Proposition

A declarative statement that is either T or F but not both

Propositional Variables

Variables that represent propositions, usually denoted by letters such as p, q, r, s

Truth Value of a Proposition

We denote the truth value of a proposition as either T or F.

Compound Propositions

Propositions formed by combining other propositions using logical operators.

Negation of p

Read as “it is not the case that p”

Conjunction ^

The proposition “p and q”, T if both p and q are T, F otherwise.

Disjunction V

The proposition “p or q”, F if p and q are both F, T otherwise.

XOR (Exclusive or)

T when exactly one of p and q is T, F otherwise

Conditional Statement / Implication

If p, then q. F when p is T and q is F, T otherwise

Converse of p → q

q → p

Inverse of p → q

not p → not q

contrapositive of p → q

not q → not p hn

Equivalent Propositions

When 2 (compound) propositions always have the same truth values, regardless of the truth value of its propositional values

Biconditional of p and q

“p if and only if q” or “if p then q and if q then p”. T when p and q have same Truth Values, F otherwise.

Tautology

A prop that is always True, independent of the truth values of the propositional variables that make it up.

Contradiction

A prop that is always F, independent of the truth values of the propositional variables that make it up.

Contingency

A prop that is neither a tautology nor a contradiction, the truth value is contingent upon the truth values of the propositional variables that make it up.

Logical Equivalence

if p < — > is a tautology.