# Discrete Mathematics

Truth Value: Either true or false. Statement: Collection of words with defined truth values. Not a question. Open Sentence: Statement which is true or false based on inputs.

EITHER: $P \vee Q$ AND: $P \wedge Q$ NOT: $\sim P$ XOR: $P \oplus Q$

Theorem 1.22—$\vee$ and $\wedge$ are commutative: $P\vee Q\equiv Q\vee P$ and $P\wedge Q\equiv Q\wedge P$.

Chartrand, Gary; Zhang, Ping. Discrete Mathematics (Page 18). Waveland Pr Inc. Kindle Edition.

##### De Morgan's Laws

$\sim (P \vee Q) = (\sim P) \wedge (\sim Q)$ $\sim (P \wedge Q) = (\sim P) \vee (\sim Q)$

### Implication

If $P$ then $Q$: $P \Rightarrow Q$ Converse: $Q \Rightarrow P$ Counterpositive: $(\sim Q)\Rightarrow (\sim P)$ Theorem 1.48—An implication is equivalent to its counterpositive: $P \Rightarrow Q \equiv (\sim Q)\Rightarrow (\sim P)$ Theorem 1.48: $P\Rightarrow Q\equiv(\sim P)\wedge Q$. Theorem 1.50: $\sim(P\Rightarrow Q)\equiv P \wedge (\sim Q)$.