Logical Equivalance

LOGICAL EQUIVALANCE : Compound propositions that have the same truth values in all possible cases are called logically equivalent.

Definition : The compound propositions P and Q are called logically equivalent if P⇔Q is a tautology. The notation P ≡ Q denotes that P and Q are logically equivalent. Some equivalance are useful for deducing other equivalance. The following table shows some important equivalance.

Logical Identities or Laws of Logic :

Note that while taking negation of compound statement ‘every’ or ‘All’ is interchanged by ‘some’ & ‘there exists’ is interchanged by ‘at least one’ & vice versa.
Example : If P : “This book is good.”
Q : “This book is costly.”
Write the following statements in symbolic form.
a) This book is good & costly.
b) This book is not good but costly.
c) This book is cheap but good.
d) This book is neither good nor costly.
e) If this book is good then it is costly.
Answers :
a) P ∧Q
b) ¬P ∧Q
c) ¬ Q∧¬  P
d) ¬ P ∧¬ Q
e) P∧Q