Notes On Implications - CBSE Class 11 Maths
Implication: A compound statement formed using the connecting words "If - then..." is called an implication or a conditional statement. Now, the compound statement can be expressed as "if p, then q". If statement p is true, then statement q will be true. Such compound statements are called implications or conditional statements. It is formed using the connecting words "if-then." p implies q is represented as p ⇒ q. A compound statement, "if p, then q," is always true, except in the case when p is true and q is false, because a true statement cannot imply a false statement. Consider the statement 'If 3 is a factor of 9, then it is also a factor of 18.' The component statements are: p: 3 is a factor of 9. q: 3 is a factor of 18. Statement p ⇒ q : 3 is a factor of 9 implies that 3 is also a factor of 18. p ⇒ q can also be expressed as: • If p, then q. • ~q  if ~p. • p only if q. • p is necessary condition for q. • q is sufficient condition for p. Converse, inverse and contrapositive of an implication Consider an implication, p ⇒ q The converse of p ⇒ q is q ⇒ p. The inverse of p ⇒ q is ~p ⇒ ~q The contrapositive of p ⇒ q is ~q ⇒ ~p. Examples: 1) Converse, inverse and contrapositive of the implication — if two sides of a triangle are equal, then it is an isosceles triangle. The component statements are: p: Two sides of a triangle are equal. q: The triangle is isosceles. The implication of the statements is p ⇒ q. The conditional statement, p ⇒ q: If two sides of a triangle are equal, then it is an isosceles triangle. Converse: q ⇒ p: If a triangle is isosceles, then two of its sides are equal. Inverse: ~p ⇒ ~q: If two sides of a triangle are not equal, then it is not an isosceles triangle. Contrapositive: ~q ⇒ ~p: If a triangle is not isosceles, then two of its sides are not equal. 2) Implication: If the sides of a triangle are equal, then its angles are also equal. Converse: If the angles of a triangle are equal, then its sides are also equal. Here, both the statements are true. Such statements can also be expressed as — the sides of a triangle are equal if and only if its angles are equal. Such compound statements are called a bi-implication or bi-conditional. Bi-implication: A compound statement involving statements p and q, which is in the form of "p if and only if q," is called a bi-implication or bi-conditional. p if and only if q is represented by p ⇒ q. p if and only if q or p ⇒ q can also be expressed as: • q if and only if p • p iff q • p is a necessary and sufficient condition for, q and vice-versa If p and q have the same truth values (either true or false), then the bi-implication is true. If either p or q is false, then the bi-implication is false.

#### Summary

Implication: A compound statement formed using the connecting words "If - then..." is called an implication or a conditional statement. Now, the compound statement can be expressed as "if p, then q". If statement p is true, then statement q will be true. Such compound statements are called implications or conditional statements. It is formed using the connecting words "if-then." p implies q is represented as p ⇒ q. A compound statement, "if p, then q," is always true, except in the case when p is true and q is false, because a true statement cannot imply a false statement. Consider the statement 'If 3 is a factor of 9, then it is also a factor of 18.' The component statements are: p: 3 is a factor of 9. q: 3 is a factor of 18. Statement p ⇒ q : 3 is a factor of 9 implies that 3 is also a factor of 18. p ⇒ q can also be expressed as: • If p, then q. • ~q  if ~p. • p only if q. • p is necessary condition for q. • q is sufficient condition for p. Converse, inverse and contrapositive of an implication Consider an implication, p ⇒ q The converse of p ⇒ q is q ⇒ p. The inverse of p ⇒ q is ~p ⇒ ~q The contrapositive of p ⇒ q is ~q ⇒ ~p. Examples: 1) Converse, inverse and contrapositive of the implication — if two sides of a triangle are equal, then it is an isosceles triangle. The component statements are: p: Two sides of a triangle are equal. q: The triangle is isosceles. The implication of the statements is p ⇒ q. The conditional statement, p ⇒ q: If two sides of a triangle are equal, then it is an isosceles triangle. Converse: q ⇒ p: If a triangle is isosceles, then two of its sides are equal. Inverse: ~p ⇒ ~q: If two sides of a triangle are not equal, then it is not an isosceles triangle. Contrapositive: ~q ⇒ ~p: If a triangle is not isosceles, then two of its sides are not equal. 2) Implication: If the sides of a triangle are equal, then its angles are also equal. Converse: If the angles of a triangle are equal, then its sides are also equal. Here, both the statements are true. Such statements can also be expressed as — the sides of a triangle are equal if and only if its angles are equal. Such compound statements are called a bi-implication or bi-conditional. Bi-implication: A compound statement involving statements p and q, which is in the form of "p if and only if q," is called a bi-implication or bi-conditional. p if and only if q is represented by p ⇒ q. p if and only if q or p ⇒ q can also be expressed as: • q if and only if p • p iff q • p is a necessary and sufficient condition for, q and vice-versa If p and q have the same truth values (either true or false), then the bi-implication is true. If either p or q is false, then the bi-implication is false.

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