Conditional Statement Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Conditional Statement.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A conditional $P \to Q$ is a statement meaning "if $P$ is true, then $Q$ must be true," read as "if $P$ then $Q$ ."

A promise or rule: if the condition holds, the consequence follows.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A conditional if P then Q is false only when P is true but Q is false.

Common stuck point: The procedure for conditional statement is the easy part; the trap is calling $P \to Q$ false when $P$ is false. Asking "Is the claim broken only when the hypothesis is true yet the conclusion is false?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the claim broken only when the hypothesis is true yet the conclusion is false?

Worked Examples

Example 1

easy

Write the converse, inverse, and contrapositive of: 'If a number is divisible by 6, then it is divisible by 3.'

Answer

\text{Contrapositive: If not divisible by 3, then not divisible by 6 (true).}

First step

Let

p

: 'a number is divisible by 6' and

q

: 'it is divisible by 3.' The original is

p \Rightarrow q

Full solution

2
Converse ( $q \Rightarrow p$ ): 'If a number is divisible by 3, then it is divisible by 6.' (False; e.g., 9.)
3
Inverse ( $\neg p \Rightarrow \neg q$ ): 'If a number is not divisible by 6, then it is not divisible by 3.' (False; e.g., 9.)
4
Contrapositive ( $\neg q \Rightarrow \neg p$ ): 'If a number is not divisible by 3, then it is not divisible by 6.' (True.)

A conditional

p \Rightarrow q

is logically equivalent to its contrapositive

\neg q \Rightarrow \neg p

, but not necessarily to its converse or inverse.

Example 2

medium

Determine the truth value of: 'If

2 > 5

, then

10 > 3

Example 3

medium

Given 'if it rains, the picnic is canceled' is true, and the picnic was NOT canceled, what can you conclude?

Example 4

hard

Translate to a conditional and find its contrapositive: 'No prime greater than

2

is even.'

Example 5

challenge

Prove using a truth table that

P \to Q

and

\neg Q \to \neg P

are logically equivalent.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Identify the hypothesis and conclusion in: 'If it rains, then the ground is wet.'

Example 2

medium

Identify the hypothesis and the conclusion in the statement: 'If a number is divisible by 4, then it is even.'

Example 3

easy

Write 'if it rains, then the ground is wet' as a conditional

P \to Q

. Identify

P

and

Q

Example 4

easy

For the conditional

P \to Q

with

P

true and

Q

true, what is its truth value?

Example 5

easy

For

P \to Q

with

P

true and

Q

false, what is its truth value?

Example 6

easy

For

P \to Q

with

P

false and

Q

true, what is its truth value?

Example 7

easy

For

P \to Q

with

P

false and

Q

false, what is its truth value?

Example 8

easy

Write the converse of '

P \to Q

Example 9

easy

Rewrite 'every multiple of 4 is even' as an if-then conditional.

Example 10

easy

In '

P \to Q

', is

P

the hypothesis or the conclusion?

Example 11

medium

If '

P \to Q

' is true and

P

is true, what can you conclude about

Q

Example 12

medium

If '

P \to Q

' is true and

Q

is true, can you conclude

P

is true?

Example 13

medium

If '

P \to Q

' is true and

Q

is false, what can you conclude about

P

Example 14

medium

Give a counterexample showing the converse of 'if

n

is a multiple of 4, then

n

is even' is false.

Example 15

medium

The statement 'if a number is divisible by 6, then it is divisible by 3' is true. Is its converse true?

Example 16

medium

Express 'you may enter only if you have a ticket' as a conditional

P \to Q

Example 17

medium

For statements

P, Q

, in how many of the four truth-value rows is

P \to Q

true?

Example 18

challenge

Prove that

P \to Q

is logically equivalent to

\neg P \lor Q

using truth values.

Example 19

challenge

Show that

(P \to Q) \land (Q \to P)

is true exactly when

P

and

Q

have the same truth value (this is the biconditional).

Example 20

challenge

A true conditional 'if a quadrilateral is a square, then it has four right angles' is given. Using the contrapositive, what can you conclude about a quadrilateral lacking four right angles?

Example 21

medium

Rewrite 'no odd number is divisible by 2' as a conditional.

Example 22

medium

Rewrite 'a number is prime is sufficient for it to be a positive integer' as

P \to Q

Example 23

easy

Write 'if a polygon is a square, then it has four equal sides' as

P \to Q

. Identify

P

and

Q

Example 24

easy

What is the truth value of 'if

1+1=3

, then the moon is cheese'?

Example 25

easy

Is the converse of a conditional always logically equivalent to the original? Answer

1

for yes,

0

for no.

Example 26

easy

Rewrite 'all squares are rectangles' as a conditional.

Example 27

easy

Is the conditional 'if

x=3

, then

x^2=9

' true for all real

x

Example 28

medium

Give a counterexample to: 'If

n^2 > 0

, then

n > 0

Example 29

medium

Express 'a positive number is necessary for the log to be defined' as a conditional.

Example 30

medium

Express 'being divisible by

4

is sufficient for being even' as a conditional.

Example 31

medium

The contrapositive of 'if

x > 0

, then

x^3 > 0

' is true. Is the original conditional also true?

Example 32

medium

Write the contrapositive of 'if

n

is even, then

n+1

is odd.'

Example 33

medium

'If a triangle has three equal sides, it is equilateral.' Write the biconditional that combines this with its true converse.

Example 34

medium

Rewrite 'a polygon is a triangle only if it has three sides' as

P \to Q

Example 35

medium

Given

P

is true and

P \to Q

is true. By which rule of inference can we conclude

Q

Example 36

hard

P \to Q

is true and

Q \to R

is true. By hypothetical syllogism, what conditional follows?

Example 37

hard

Negate the conditional

P \to Q

Example 38

hard

Determine the truth value: 'If

\sqrt{2}

is rational, then

\pi

is rational.'

Example 39

hard

Affirming the consequent is invalid. Given

P \to Q

true and

Q

true, can we conclude

P

? Answer

1

for yes,

0

for no.

Example 40

hard

Build a truth table for

(P \to Q) \land (Q \to P)

and count how many rows are true.

Example 41

challenge

Find a quadrilateral that is a counterexample to the converse of: 'If a quadrilateral is a rhombus, then its diagonals are perpendicular.'

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Related Concepts

Logical Statement Contrapositive Biconditional

Background Knowledge

These ideas may be useful before you work through the harder examples.

logical statement