Practice Proofs in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick 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.
Showing a random 20 of 50 problems.
Example 1
challengeProve: for any sets, if and only if .
Example 2
hardProve by induction: for all .
Example 3
easyA proof by contradiction begins by assuming what?
Example 4
mediumProve: if is rational and is irrational, then is irrational.
Example 5
easyProve directly: The sum of two even integers is even.
Example 6
mediumProve: if and are both odd, then is odd.
Example 7
mediumProve by contradiction: there is no largest integer.
Example 8
mediumProve: for all integers , is even.
Example 9
mediumProve: if , then for every integer .
Example 10
easyProve directly: if is even, then is even.
Example 11
mediumProve: if is even, then is even (use contrapositive).
Example 12
mediumProve by contrapositive: if is even, then is even.
Example 13
easyProve: the sum of two odd numbers is even.
Example 14
mediumDisprove by counterexample: 'For all integers , .'
Example 15
mediumProve: the sum of the first odd numbers equals — outline the method that fits best.
Example 16
mediumWhat is the converse of 'if it rains, the ground is wet', and is it always true?
Example 17
hardProve: there are infinitely many primes (Euclid's argument).
Example 18
easyWhat name do we give a statement that has been proven from axioms?
Example 19
easyWhat proof technique uses 'base case' and 'inductive step'?
Example 20
mediumProve directly: For any integer , if is odd then is odd.