Logical-equivalencies.md

Logical Equivalencies

Preface:

The compound propositions $$p$$ and $$q$$ are said to be logically equivalent if $$p \iff q$$ is a tautology (i.e., all their respective enteries in their truth table are congruent). Example, the compound proposition $$p \iff q$$ is logically equivalent to $$(p \implies q) \land (q \implies p)$$; because both the $$p \implies q$$ and $$q \implies p$$ have the same values in their respective enteries, and $$(p \iff q) \iff ((p \implies q) \land (q \implies p))$$ is a tautology.

Note

• A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it, is called a tautology.
• A compound proposition that is always false, no matter what the truth values of the propositional variables that occur in it, is called a contradication.
• A compound proposition that is neither a tautology nor a contradiction, is a contingency.

1) Exclusive OR Statement:

2) Implication Statement:

• Preface: A propositional statement utilized as a conditional logical statement.
• Statement: $$(p \implies q) \equiv (\neg p \vee q) \equiv (\neg (p \land \neg q))$$
• Truth table:

p	q	$$p \implies q$$	$$\neg p \vee q$$	$$\neg (p \land \neg q)$$
T	T	T	T	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

• Truth table for the Bi-conditional compound propositions:

$$(p \implies q) \implies (\neg p \vee q)$$	$$(\neg p \vee q) \implies (p \implies q)$$	$$(p \implies q) \iff (\neg p \vee q)$$
T	T	T
T	T	T
T	T	T
T	T	T

• Since, $$(p \implies q) \iff (\neg p \vee q)$$ is a tautology; then $$(p \implies q) \equiv (\neg p \vee q)$$ holds!
• The same applies with the second logical equivalency.

3) Bi-conditional Statement: