Propositional Variables

Each propositional variable stands for a proposition, something that is either true or false.
- convention is lower-case letters, such as $p,q,r,s$
- each variable is like a bool variable

Propositional Connectives

Propositional connectives link propositions into larger propositions (e.g. $\lor$, $\land$, $\to$, etc.)
Seven propositional connectives, five of which will be familiar from programming
- Notes: $\lor$ is inclusive or
- true and false are considered connectives that connect 0 things
- Exclusive or: can be written as $(p \lor q) \land \lnot (p \land q)$, or $(\lnot p \land q) \lor (p \land \lnot q)$
The convention for proofs is NOT to use propositional logic notation, and to instead use words. Exception: if proof about the actual logic notation or computational logic.

Written	Read	Fancy name	C++
$\lnot p$	not p	negation	`!p`
$p \land q$	p and q	conjunction	`p && q`
$p \lor q$	p or q	disjunction	`p
$\top$	true	truth	`true`
$\bot$	false	falsity	`false`
$p \to q$	p implies q	material conditional	???
$p \leftrightarrow q$	p if and only if q	biconditional	???

$\lnot (p \land q) = \lnot p \lor \lnot q$
$\lnot(p \lor q) = \lnot p \land \lnot q$
pro tip in code: use de Morgan’s laws to do short-circuit evaluation. Prefer right side over left side in above equations for code.

Contrapositive in a new light: $p \to q = \lnot q \to \lnot p$
Another thing: you can assume $p$ is true to prove an implication, since if $p$ is false, then $p \to q$ is trivially true.
Proof by contradiction: $(\lnot p \to \bot) \to p$