Variables

$\land$
→
and
$\lor$
→
or
$\lnot$
→
not
$\iff$
→
if and only if
1.
Conjunction
$\begin{matrix} p & q & p \land q \\[0.3 em] T & T & T \\[0.3 em] T & F & F \\[0.3 em] F & T & F \\[0.3 em] F & F & F \end{matrix}$
Logic
2.
Disjunction
$\begin{matrix} p & q & p \lor q \\[0.3 em] T & T & T \\[0.3 em] T & F & T \\[0.3 em] F & T & T \\[0.3 em] F & F & F \end{matrix}$
Logic
3.
Implication
$\begin{matrix} p & q & p \implies q \\[0.3 em] T & T & T \\[0.3 em] T & F & F \\[0.3 em] F & T & F \\[0.3 em] F & F & T \end{matrix}$
Logic
4.
Law of noncontradiction
$p \land \lnot p = F$
Logic
5.
Double Negation
$\lnot \left(\lnot p \right) \iff p$
Logic
6.
Commutativity for conjunction
$p \land q \iff q \land p$
Logic
7.
Commutativity for disjunction
$p \lor q \iff q \lor p$
Logic
8.
Associativity for conjunction
$\left(p \land q\right) \land r \iff p \land \left(q \land r\right)$
Logic
9.
Associativity for disjunction
$\left(p \lor q\right) \lor r \iff p \lor \left(q \lor r\right)$
Logic
10.
Conjunction idempotence
$p \land p \iff p$
Logic
11.
Disjunction idempotence
$p \lor p \iff p$
Logic
12.
Distributive Property
$p \land ( q \lor r) \iff (p \land q) \lor (p \land r)$
Logic
13.
Distributive Property
$p \lor ( q \land r) \iff (p \lor q) \land (p \lor r)$
Logic
14.
Implication is transitive
$(p \implies q) \land (q \implies r) \implies p \implies r$
Logic
15.
Negation of implication
$\lnot (p \implies r) \iff p \land \lnot q$
Logic
16.
Contrapositive of an Implication
$(p \implies q) \iff (\lnot q \implies \lnot p)$
Logic
17.
Equivalence as double implication
$(p \iff q) \iff \left[(p \implies q) \land (q \implies p)\right]$
Logic
18.
Equivalence is transitive
$(p \iff q) \land (q \iff r) \iff (p \iff r)$
Logic
19.
De Morgan's laws for conjunction
$\lnot ( p \land q) \iff \lnot p \lor \lnot q$
Logic
20.
De Morgan's laws for disjunction
$\lnot (p \lor q) \iff \lnot p \land \lnot q$
Logic

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