# Conditional and BiConditional Statements

## Conditional Statement

Let p and q are two statements then "if p then q" is a compound statement, denoted by p→ q and referred as a conditional statement, or implication. The implication p→ q is false only when p is true, and q is false; otherwise, it is always true. In this implication, p is called the hypothesis (or antecedent) and q is called the conclusion (or consequent).

 p q p → q T T T T F F F T T F F T

For Example: The followings are conditional statements.

1. If a = b and b = c, then a = c.
2. If I get money, then I will purchase a computer.

## Variations in Conditional Statement

Contrapositive: The proposition ~q→~p is called contrapositive of p →q.

Converse: The proposition q→p is called the converse of p →q.

Inverse: The proposition ~p→~q is called the inverse of p →q.

Example1: Show that p →q and its contrapositive ~q→~p are logically equivalent.

Solution: Construct the truth table for both the propositions:

 p q ~p ~q p →q ~q→~p T T F F T T T F F T F F F T T F T T F F T T T T

As, the values in both cases are same, hence both propositions are equivalent.

Example2: Show that proposition q→p, and ~p→~q is not equivalent to p →q.

Solution: Construct the truth table for all the above propositions:

 p q ~p ~q p →q q→p ~p→~q T T F F T T T T F F T F T T F T T F T F F F F T T T T T

As, the values of p →q in a table is not equal to q→p and ~p→~q as in fig. So both of them are not equal to p →q, but they are themselves logically equivalent.

## BiConditional Statement

If p and q are two statements then "p if and only if q" is a compound statement, denoted as p ↔ q and referred as a biconditional statement or an equivalence. The equivalence p ↔ q is true only when both p and q are true or when both p and q are false.

 p q p ↔ q T T T T F F F T F F F T

For Example: (i) Two lines are parallel if and only if they have the same slope.
(ii) You will pass the exam if and only if you will work hard.

Example: Prove that p ↔ q is equivalent to (p →q) ∧(q→p).

Solution: Construct the truth table for both the propositions:

 p q p ↔ q T T T T F F F T F F F T

 p q p →q q→p (p →q)∧(q→p) T T T T T T F F T F F T T F F F F T T T

Since, the truth tables are the same, hence they are logically equivalent. Hence Proved.

## Principle of Duality

Two formulas A1 and A2 are said to be duals of each other if either one can be obtained from the other by replacing ∧ (AND) by ∨ (OR) by ∧ (AND). Also if the formula contains T (True) or F (False), then we replace T by F and F by T to obtain the dual.

## Equivalence of Propositions

Two propositions are said to be logically equivalent if they have exactly the same truth values under all circumstances.

The table1 contains the fundamental logical equivalent expressions:

Laws of the algebra of propositions

 Idempotent laws (i) p ∨ p≅p (ii) p ∧ p≅p Associative laws (i) (p ∨ q) ∨ r ≅ p∨ (q ∨ r) (ii) (p ∧ q) ∧ r ≅ p ∧ (q ∧ r) Commutative laws (i) p ∨ q ≅ q ∨ p (ii) p ∧ q ≅ q ∧ p Distributive laws (i) p ∨ (q ∧ r) ≅ (p ∨ q) ∧ (p ∨ r) (ii) p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r) Identity laws (i)p ∨ F ≅ p (iv) p ∧ F≅F (ii) p ∧ T≅ p (iii) p ∨ T ≅ T Involution laws (i) ¬¬p ≅ p Complement laws (i) p ∨ ¬p ≅ T (ii) p ∧ ¬p ≅ T DeMorgan's laws: (i) ¬(p ∨ q) ≅ ¬p ∧ ¬q (ii) ¬(p ∧ q) ≅¬p ∨ ¬q

Example: Consider the following propositions

Are they equivalent?

Solution: Construct the truth table for both

 p q ~p ~q ~p∨∼q p∧q ~(p∧q) T T F F F T F T F F T T F T F T T F T F T F F T T T F T   