Proofs Examples: 46 Problems with Answers

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

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 mathematical proof is a rigorous logical argument that demonstrates the truth of a statement beyond doubt, proceeding from accepted axioms and previously proven results through valid inference rules.

It is not guessing the answer; it is proving why the answer must be true.

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 proof is a rigorous chain of valid inferences from accepted axioms to a guaranteed conclusion.

Common stuck point: The procedure for proofs is the easy part; the trap is treating confirming examples as a proof. Asking "Does my argument guarantee the claim for EVERY case via valid inference, not just confirm examples?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does my argument guarantee the claim for EVERY case via valid inference, not just confirm examples?

Worked Examples

Example 1

easy

Prove directly: The sum of two even integers is even.

Answer

a + b = 2 (m + n) is even.

First step

Let

a

and

b

be even integers. By definition,

a = 2m

and

b = 2n

for some integers

m, n

Full solution

2
Then $a + b = 2m + 2n = 2(m + n)$ .
3
Since $m + n$ is an integer, $a + b = 2(m + n)$ is even by definition.

A direct proof starts from the hypothesis, applies definitions and algebra, and arrives at the conclusion. Translating 'even' into

2k

is a standard first step.

Example 2

medium

Prove directly: For any integer

n

, if

n

is odd then

n^2

is odd.

Example 3

medium

Prove by contradiction: there is no largest integer.

Example 4

medium

Prove by contrapositive: if

n^2

is even, then

n

is even.

Example 5

medium

Prove: if

m

and

n

are both perfect squares, then

mn

is a perfect square.

Example 6

hard

Prove by contradiction that

\sqrt{2}

is irrational.

Example 7

medium

Prove by induction:

1+2+\dots+n = \frac{n(n+1)}{2}

Example 8

medium

Prove: if

a \mid b

, then

a \mid bc

for every integer

c

Example 9

hard

Prove by cases: for any integer

n

n^2

has remainder

0

1

when divided by

4

Example 10

medium

Prove: the sum of three consecutive integers is divisible by

3

Example 11

hard

Prove by induction:

2^n > n

for all

n \ge 1

Example 12

medium

Prove: if

a, b

are rational, then

a+b

is rational.

Example 13

hard

Prove: there are infinitely many primes (Euclid's argument).

Example 14

challenge

Prove: for any natural number

n

n^3 - n

is divisible by

6

Practice Problems

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

Example 1

medium

Prove: The product of two odd integers is odd.

Example 2

medium

Prove directly: if

n

is even, then

n+1

is odd.

Example 3

easy

In a proof, what are the two things you must clearly identify before reasoning?

Example 4

easy

Prove directly: if

n

is even, then

n+2

is even.

Example 5

easy

What is wrong with proving '

x=2

' by writing 'assume

x=2

, then

x=2

Example 6

easy

Prove: the sum of two odd numbers is even.

Example 7

easy

Which is a valid proof of '

\forall n,\ 2n

is even': (A) test

n=1,2,3

; (B) write

2n=2\cdot n

Example 8

easy

In a proof, what role does an axiom play?

Example 9

easy

Prove: if

a \mid b

and

a \mid c

, then

a \mid (b+c)

Example 10

easy

What does 'it is obvious' hide in a proof?

Example 11

medium

Prove: if

n^2

is even, then

n

is even (use contrapositive).

Example 12

medium

Prove: for all integers

n

n^2+n

is even.

Example 13

medium

Prove: if

x

is rational and

y

is irrational, then

x+y

is irrational.

Example 14

medium

What is the converse of 'if it rains, the ground is wet', and is it always true?

Example 15

medium

Prove: the sum of the first

n

odd numbers equals

n^2

— outline the method that fits best.

Example 16

medium

Identify the flaw: 'Let

a=b

. Then

a^2=ab

a^2-b^2=ab-b^2

(a+b)(a-b)=b(a-b)

, so

a+b=b

, thus

2b=b

Example 17

medium

Prove: if

a < b

then

a < \frac{a+b}{2} < b

Example 18

medium

State what makes a proof 'valid' versus merely 'persuasive'.

Example 19

medium

Prove: if

a

and

b

are both odd, then

ab

is odd.

Example 20

challenge

Prove: there are infinitely many primes (outline Euclid's argument).

Example 21

challenge

Prove by strong induction: every integer

n \ge 2

has a prime factor.

Example 22

challenge

Prove: for any sets,

A \subseteq B

if and only if

A \cap B = A

Example 23

easy

In a direct proof of 'if

P

then

Q

', what do you assume at the start?

Example 24

easy

What is one counterexample sufficient to disprove?

Example 25

medium

Disprove by counterexample: 'For all integers

n

n^2 > n

Example 26

easy

True or false: a proof by example (showing the statement holds for

n=1,2,3

) proves the universal statement

\forall n

Example 27

medium

Prove: if

n

is an integer, then

n^2 + n

is even.

Example 28

medium

Prove: an integer

n

is even if and only if

n+2

is even.

Example 29

easy

What name do we give a statement that has been proven from axioms?

Example 30

medium

Identify the flaw: 'Proof that all horses are the same color: by induction on the size of a group of horses.'

Example 31

easy

True or false: a single example proves a universal claim.

Example 32

medium

Disprove: 'For all primes

p

2^p - 1

is prime.'

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Related Concepts

Logical Statement Conditional Statement Proof (Intuition)

Background Knowledge

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

logical statementconditionalproof intuition