Biconditional Statements(\iff, \Leftrightarrow, \equiv, \leftrightarrow)
"P is and only if Q"
"P is logically equivalent to Q"
(P \implies Q)\land(Q\implies P)
P | Q | P->Q | Q->P | P<->Q |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | T | F |
F | T | T | F | F |
F | F | T | T | T |
DeMorgan's Laws:
tautology: a proposition that is always true for all possible truth assignments.
DeMorgan's Law for Connectives: "For all propositions P and Q, the following logical equivalences holds true"
DeMorgan's Laws for Sets: "For any sets A and B with a universal set U, the following hold true."
P | Q | P \land Q | \lnot(P \land Q) | \lnot P \lor \lnot Q | \lnot(P\land Q) \iff \lnot P \lor \lnot Q |
---|---|---|---|---|---|
T | T | T | F | F | T |
T | F | F | T | T | T |
F | T | F | T | T | T |
F | F | F | T | T | T |
Distributive Law for Connectives:
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)
you can think this as distributive
Law of Double Negation: \lnot \lnot P \equiv P
Disjunctive Form of Implication: P\implies Q\equiv \lnot P \lor Q
Disjunctive Form of Implication: P\iff Q \equiv (P\implies Q)\land (Q\implies P)
Contraposition(contrapositive): P\implies Q \equiv \lnot Q \implies P
converse: Q\implies P is converse of P\implies Q
inverse: \lnot P \implies \lnot Q is inverse of P\implies Q
converse and inverse are logically equivalent to each other (since they are contraposition to each other)
Law of Excluded Middle: P(x) \neq \lnot P(x)
both our qualifiers changed
right (inner) most variable proposition negated
Maximally Negated Form:
\lnot(\forall x \in S)P(x)\equiv(\exists x \in S)\lnot P(x)
\lnot(\exists x \in S)P(x)\equiv(\forall x \in S)\lnot P(x)
Notice that S, P(x), Q(y) does not change
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