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## Definition of Rational

A real number r is rational if, and only if, it can be expressed as a quotient of two integers with a non-zero denominator.

$r~is~rational\iff\exists~integers~a~and~b~such~that~r=\frac{a}{b}~and~b\neq 0$

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## Definition of Even

An integer n is even if, and only if, n equals twice some integer.

$n~is~even\iff\exists~an~integer~k~such~that~n=2k$

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## Rules of Inference

A rule of inference is a form of argument that is valid.

 Modus Ponens $p \rightarrow q \newline p \newline \therefore q$$p \rightarrow q \newline p \newline \therefore q$ Modus Tollens $p \rightarrow q \newline \neg q \newline \therefore \neg p$$p \rightarrow q \newline \neg q \newline \therefore \neg p$ Conjunction $p \newline q \newline \therefore p \land q$$p \newline q \newline \therefore p \land q$ Contradiction $\neg p \rightarrow c \newline \therefore p$$\neg p \rightarrow c \newline \therefore p$ Transitivity $p \rightarrow q \newline q \rightarrow r \newline \therefore p \rightarrow r$$p \rightarrow q \newline q \rightarrow r \newline \therefore p \rightarrow r$ Proof by Division into Cases $p \lor q \newline p \rightarrow r \newline q \rightarrow r \newline \therefore r$$p \lor q \newline p \rightarrow r \newline q \rightarrow r \newline \therefore r$ Generalization $p \newline \therefore p \lor q$$p \newline \therefore p \lor q$ $q \newline \therefore q \lor p$$q \newline \therefore q \lor p$ Specialization $p \land q \newline \therefore p$$p \land q \newline \therefore p$ $p \land q \newline \therefore q$$p \land q \newline \therefore q$ Elimination $p \lor q \newline \neg q \newline \therefore p$$p \lor q \newline \neg q \newline \therefore p$ $p \lor q \newline \neg p \newline \therefore q$$p \lor q \newline \neg p \newline \therefore q$
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## Logical Equivalences

Given any statement variables p, q, and r, a tautology t and a contradiction c, the following logical equivalences hold.

 Commutative laws $p \land q \equiv q \land p$$p \land q \equiv q \land p$ $p \lor q \equiv q \lor p$$p \lor q \equiv q \lor p$ Associative laws $(p \land q) \land r \equiv p \land (q \land r)$$(p \land q) \land r \equiv p \land (q \land r)$ $(p \lor q) \lor r \equiv p \lor (q \lor r)$$(p \lor q) \lor r \equiv p \lor (q \lor r)$ Distributive laws $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$ $p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$$p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$ Idempotent laws $p \land p \equiv p$$p \land p \equiv p$ $p \lor p \equiv p$$p \lor p \equiv p$ Absorption laws $p \lor (p \land q) \equiv p$$p \lor (p \land q) \equiv p$ $p \land (p \lor q) \equiv p$$p \land (p \lor q) \equiv p$ De Morgan’s laws $\neg (p \land q) \equiv \neg p \lor \neg q$$\neg (p \land q) \equiv \neg p \lor \neg q$ $\neg (p \lor q) \equiv \neg p \land \neg q$$\neg (p \lor q) \equiv \neg p \land \neg q$ Identity laws $p \land t \equiv p$$p \land t \equiv p$ $p \lor c \equiv p$$p \lor c \equiv p$ Universal Bound laws $p \lor t \equiv p$$p \lor t \equiv p$ $p \land c \equiv c$$p \land c \equiv c$ Negation laws $p \lor \neg p \equiv t$$p \lor \neg p \equiv t$ $p \land \neg p \equiv c$$p \land \neg p \equiv c$ Double Negative law $\neg (\neg p) \equiv p$$\neg (\neg p) \equiv p$ Negations of t and c $\neg t \equiv c$$\neg t \equiv c$ $\neg c \equiv t$$\neg c \equiv t$
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## Biconditional

$p \leftrightarrow q$

Given statement variables p and q, the biconditional of p and q is “p if, and only if, q.” If both p and q have the same truth values the statement is true, and it is false if p and q have opposite truth values.

$\displaystyle p$

$\displaystyle q$

$\displaystyle p \leftrightarrow q$

T T T
T F F
F T F
F F T