Practice Proof (Intuition) in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The informal, intuitive sense of why a mathematical statement must be true — the "aha" that precedes and motivates a formal proof.

A chain of reasoning that convinces you something MUST be true.

Showing a random 20 of 50 problems.

Example 1

easy
Build intuition for the statement: 'For any integer nn, n(n+1)n(n+1) is even.' Explain informally why this must be true.

Example 2

medium
In the inductive step for ∑i=1ni=n(n+1)2\sum_{i=1}^{n} i=\frac{n(n+1)}{2}, assuming it for n=kn=k, what do you add to both sides to reach n=k+1n=k+1? Give the term.

Example 3

easy
The contrapositive of 'if PP then QQ' is 'if not QQ then not PP' and is logically equivalent. Give 11 if equivalent.

Example 4

medium
Build intuition for why 2\sqrt{2} is irrational before writing the formal proof. What is the core contradiction?

Example 5

hard
Why does the principle of strong induction allow assuming P(1),P(2),…,P(k)P(1), P(2), \ldots, P(k) all at once when proving P(k+1)P(k+1)? Build intuition with the prime factorization theorem.

Example 6

easy
How many cases does a proof by cases on the parity of an integer need? Give the number.

Example 7

easy
How many counterexamples are needed to disprove a universal statement of the form 'for all xx, P(x)P(x)'?

Example 8

easy
The proof technique that assumes the opposite and derives a contradiction is called proof by _____.

Example 9

easy
The sum of two even numbers is even. Writing 2a+2b=2(a+b)2a+2b=2(a+b), what common factor proves it? Give the factor.

Example 10

challenge
In a flawed proof '1=21=2', both sides are divided by (a−b)(a-b) after setting a=ba=b. What is the value of a−ba-b that invalidates this step?

Example 11

medium
Give a counterexample to the claim 'every odd integer is prime'.

Example 12

hard
If 'all crows are black' is true and you see a non-black object, what can you conclude about the object?

Example 13

medium
Sketch the intuition behind the pigeonhole principle: if n+1n + 1 pigeons fit into nn holes, what must happen?

Example 14

challenge
A proof shows n3−nn^3-n is divisible by 33 for all integers nn. Factoring gives (n−1)n(n+1)(n-1)n(n+1) — how many consecutive integers is that product, guaranteeing a multiple of 33?

Example 15

medium
To prove a number is divisible by 66, it suffices to show divisibility by which two coprime numbers? Give their product.

Example 16

easy
The negation of 'for all xx, P(x)P(x)' is 'there exists xx such that _____'.

Example 17

easy
True or false: 'if PP then QQ' is logically equivalent to its contrapositive 'if not QQ then not PP'.

Example 18

medium
A valid proof needs each step to follow. In 'a=ba=b, so a2=aba^2=ab', what operation was applied to both sides? Give the multiplier.

Example 19

easy
Is the statement 'for some nn, n2=nn^2 = n' true? Give an example.

Example 20

medium
In an 'if and only if' proof, how many directions must be shown? Give the number.
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