# 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.