4.4 Implication

(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

$\mathrm{(b)\; I}fP\cup Q=P,thenQ\subset P$

Solution:

1. Antecedent: m = 2
   Consequent: 2m² + m = 10
2. $\begin{array}{l}\text{Antecedent}:P\cup Q=P\\ \text{Consequent}:Q\subset P\end{array}$
   

(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

1. If p, then q.
2. p if and only if q.

Solution:

1. If x + 1 = 8, then x = 7.
2. 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.

1. If x² + x – 2 = 0, then (x – 1)(x + 2) = 0.
2. If x = 7, then x + 2 = 9.

Solution:

1. If (x – 1)(x + 2) = 0, then x² + x – 2.
2. If x + 2 = 9, then x = 7.