Conditional and BiConditional StatementsConditional StatementLet 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).
For Example: The followings are conditional statements.
Variations in Conditional StatementContrapositive: 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:
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:
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 StatementIf 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.
For Example: (i) Two lines are parallel if and only if they have the same slope. Example: Prove that p ↔ q is equivalent to (p →q) ∧(q→p). Solution: Construct the truth table for both the propositions:
Since, the truth tables are the same, hence they are logically equivalent. Hence Proved. Principle of DualityTwo formulas A_{1} and A_{2} 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. Note1: The two connectives ∧ and ∨ are called dual of each other.

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 