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Conditional Statement

In the study of logic, when two statements are combined then they are known as compound statements. They can be conditional or bi-conditional. If a statement is given that if it rains we will not play. This is a combination of two statements. This statement is in the form of "If ... then" and hence is a conditional statement. A conditional statement having statements p and q can be written in given ways.

1: p implies q.
2: p is sufficient for q.
3:q is necessary for p.
4: p $\Rightarrow$ q

Let us take an example and understand. a = 6 then b = a + 2 must be 8. Hence, a $\Rightarrow$ b.

Types of Conditional Statements

Below are the listed types of conditional statements:

1. Conditional statement

2. Bi – conditional statement

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Conditional statement

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  • Conditional statement is also called as implications.

  • Form of conditional statement is if P then Q

  • Sign of logical connector conditional statement is $\rightarrow$. Example P $\rightarrow$ Q pronouns as P implies Q.

  • The state P $\rightarrow$ Q is false if the P is true and Q is false otherwise P $\rightarrow$ Q is true.

Truth table for Conditional statement:

A

B

A$\rightarrow$B

T

T

T

T

F

F

F

T

T

F

F

T

We have a conditional statement If it is raining, we will not play. Let, A: It is raining and B: we will not play.
If A is true, that is, it is raining and B is false, that is, we played, then the statement A implies B is false.
If A is false, that is, it is not raining and B is true, that is, we did not play, still the statement is true. As A is the necessary condition for B but it is not sufficient. If A is true, B should be true but if A is false B may or may not be true.

Bi-conditional statement

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A statement showing an "if and only if" relation is known as a biconditional statement. An event P will occur if and only if the event Q occurs, which means if P has occured then it implies Q will occur and vice versa.
$P\leftrightarrow Q \Rightarrow (P\rightarrow Q)\vee (Q\rightarrow P)$

Example :
P: A number is divisible by 2.
Q: A number is even.
If P will occur then Q will occur and if Q will occur then P will occur.
Hence, P will occur if and only if Q will occur.
We can say that $P\leftrightarrow Q$.

Examples

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Problem 1: If a > 0 is a positive number then a = 5 is correct or not justify your answer.

Solution:

Step 1: here a > 0 and it is a positive number

Step 2: so it is given a=5

Step 3: here first thing is correct that a is positive but

a = 5 is not correct because it can be any number greater than 0.

Problem 2:

 P                     Q                         P$\Rightarrow$     
 I am late  I am on time  
 I am normal  I am abnormal  


Justify P$\Rightarrow$

Solution: Here if statement P is correct then statement Q is not correct

and if statement Q is correct then statement P is not correct

SO here P $\Rightarrow$ Q is not correct

because both statement are not similar.

similarly in the second statement 

here if statement P is correct then statement Q is not correct

here if statement Q is correct then statement P is not correct

SO here P $\Rightarrow$ Q is not correct

because both statement are not similar.

Practice Problems

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Problem 1: For the given statements find if $P\rightarrow Q$ exists?

P: x is divisible by 2.
Q: x is divisible by 4.
Problem 2: Does $P\leftrightarrow Q$ exists for given statements?

P: x is a prime number.
Q: x is not divisible by 2.
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