4.4 Implications Short Notes

4.4 Implications

(A) Antecedent and Consequent of an Implication

1. For two statements p and q, the sentence ‘if p, then q’ is called an implication.

2. p is called the antecedent.
   q is called the consequent.

Example:
Identify the antecedent and consequent of the following implications.
(a) If m = 2, then 2m2 + m = 10
(b) If PQ=P, then QP

Solution:
(a) Antecedent: m = 2
       Consequent:2m2 + m = 10

(b) Antecedent:PQ=P Consequent:QP


(B) Implications of the Form ‘p if and only if q

1. Two implications ‘if p, then q’ and ‘if q, then p’ can be written as ‘p if and only if q’.

2. Likewise, two statements can be written from a statement in the form p if and only if q as follows:
    Implication 1: If p, then q.
    Implication 2: If q, then p.

Example 1:
Given that p: x + 1 = 8
                 q: x = 7
Construct a mathematical statement in the form of implication
(a) If p, then q.
(b) p if and only if q.

Solution:
(a) If x + 1 = 8, then x = 7.
(b) x + 1 = 8 if and only if x = 7. 


Example 2:
Write down two implications based on the following sentence:
x3 = 64 if and only if x = 4.

Solution:
If x3 = 64, then x = 4.
If x = 4, then x3 = 64.

(C) Converse of an Implication

1. The converse of an implication ‘if p, then q’ is ‘if q, then p’.

Example:
State the converse of each of the following implications.
(a) If x2 + x – 2 = 0, then (x - 1)(x + 2) = 0.
(b) If x = 7, then x + 2 = 9.

Solution:
(a) If (x - 1)(x + 2) = 0, then x2 + x – 2.
(b) If x + 2 = 9, then x = 7.