Conditional Statement

In the study of logic, there are two types of statements, conditional statement and bi-conditional statement. These statements are formed by combining two statements, which are called compound statements. Suppose a statement is- if it rains, then we don’t play. This is a combination of two statements. These types of statements are mainly used in computer programming languages such as c, c++, etc. Let us learn more here with examples.

Conditional Statement Definition

A conditional statement is represented in the form of “if…then”. Let p and q are the two statements, then statements p and q can be written as per different conditions, such as;

  • p implies q
  • p is sufficient for q
  • q is necessary for p
  • p ⇒ q

Points to remember:

  1. A conditional statement is also called implications.
  2. Sign of logical connector conditional statement is →. Example P → Q pronouns as P implies Q.
  3. The state P → Q is false if the P is true and Q is false otherwise P → Q is true.

Truth Table for Conditional Statement

The truth table for any two inputs, say A and B is given by;

A

B

A→B

T

T

T

T

F

F

F

T

T

F

F

T

Example: We have a conditional statement If it is raining, we will not play. Let, A: It is raining and B: we will not play. Then;

  • 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. 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.

What is a Bi-Conditional Statement?

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 occurred then it implies Q will occur and vice versa.

P ↔ Q ⇒ (P→Q) ∨ (Q→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↔Q.

Conditional Statement Examples

Q.1: If a > 0 is a positive number, then is a = 10 correct or not? Justify your answer.

Solution: Given, a > 0 and is a positive number

And it is given a = 10

So the first statement a > 0 is correct because any number greater than 0 is a positive number. But a = 10 is not a correct statement because it can be any number greater than 0.

Q.2: Justify P → Q, for the given table below.

P

Q

P → Q

I am late

I am on time

I am punctual

I am on time

Solution: Case 1: We can see, for the first row, in the given table,

If statement P is correct, then Q is incorrect and if Q is correct then P is incorrect. Both the statements contradict each other.

Hence, P → Q = False

Case 2: In the second row of the given table, if P is correct then Q is correct and if Q is correct then P is also correct. Hence, it satisfies the condition.

P → Q = True

Therefore, we can construct the table;

P

Q

P → Q

I am late

I am on time

F

I am punctual

I am on time

T