**4.4 Implications**

**(A) Antecedent and Consequent of an Implication**

**1.**For two statements

*and*

**p***, the sentence ‘*

**q****if**’ is called an

*p*, then*q***implication**.

**2.**

*is called the*

**p****antecedent**.

**is called the**

*q***consequent**.

Example:

Example:

Identify the antecedent and consequent of the following implications.

(a) Antecedent:

Consequent:: 2

(b) $\begin{array}{l}\text{Antecedent}:P\cup Q=P\\ \text{Consequent}:Q\subset P\end{array}$

(a) If

$\mathrm{(b)\; I}f\text{}P\cup Q=P,\text{}then\text{}Q\subset P$
*m*= 2, then 2*m*^{2}+*m*= 10

**Solution:**(a) Antecedent:

*m*= 2

Consequent:: 2

*m*

^{2}+

*m*= 10

(b) $\begin{array}{l}\text{Antecedent}:P\cup Q=P\\ \text{Consequent}:Q\subset P\end{array}$

**(B) Implications of the Form ‘**

*p*if and only if*q*’**1.**Two implications ‘

**if**’ and ‘

*p*, then*q***if**’ can be written as ‘

*q*, then*p***’.**

*p*if and only if*q***2.**Likewise, two statements can be written from a statement in the form

**‘**as follows:

*p*if and only if*q*’ 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: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:

*x*

^{3}= 64 if and only if

*x*= 4.

Solution:Solution:

If

*x*^{3}= 64, then*x*= 4.If

*x*= 4, then*x*^{3}= 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

*x*^{2}+*x*– 2 = 0, then (*x*– 1)(*x*+ 2) = 0.(b) If

*x*= 7, then*x*+ 2 = 9.

Solution:Solution:

(a) If (

*x*– 1)(*x*+ 2) = 0, then*x*^{2}+*x*– 2.(b) If

*x*+ 2 = 9, then*x*= 7.