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:
- A conditional statement is also called implication.
- The sign of the logical connector conditional statement is →. Example P → Q pronouns as P implies Q.
- 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;
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)|
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|