Conditional Statement

How to find the hypothesis, the conclusion, and the truth value of a conditional statement: definition, truth value, 6 examples, and their solutions.

Definition

Definition

[p → q] is a conditional statement.

It means [If p, then q].

p is the hypothesis.
q is the conclusion.

Example 1

Example

Solution

The structure of the given statement is
[if ... , then ...].

It's a conditional statement.

So the statement behind if is p:
2 is a prime number.

And the statement behind then is q:
2 is an odd number.

So the hypothesis, p, is
2 is a prime number.

And the conclusion, q, is
2 is an odd number.

So this is the answer.

Example 2

Example

Solution

The structure of the given statement is
[if ... , then ...].

It's a conditional statement.

So the statement behind if is p:
he is not in his room.

And the statement behind then is q:
he is playing basketball.

So the hypothesis, p, is
he is not in his room.

And the conclusion, q, is
he is playing basketball.

So this is the answer.

Example 3

Example

Solution

The structure of the given statement is
[... if ...].

[If] is in the middle of the statement.
But this is still a conditional statement.

So the statement behind if is p:
it's raining.

And q is the former statement:
I'll tell you.

So the hypothesis, p, is
it's raining.

And the conclusion, q, is
I'll tell you.

So this is the answer.

Truth Value

Truth Table

A conditional statement is false
if p is true and q is false.
(True hypothesis and false conclusion
makes a conditional false.)

Otherwise,
a conditional statment is true.

If the hypothesis, p, is false,
then p → q is true.
It doesn't matter
whether the conclusion q is true or false.

Example 4

Example

Solution

p: 2 is a prime number.

This is true.

q: 2 is a positive number.

This is also true.

Both p and q are true.

So p → q is true.

So true is the answer.

Example 5

Example

Solution

p: 2 is a prime number.

This is true.

r: 2 is an odd number.

This is false.

p is true.
r is false.

So p → r is false.

So false is the answer.

Example 6

Example

Solution

r: 2 is an odd number.

This is false.

The hypothesis r is false.

Then r → p is true.

It doesn't matter
whether the conclusion p is true or not.

So true is the answer.