Mathematical Proofs Worksheet
This worksheet covers various methods of mathematical proofs, including direct proof, proof by contradiction, and proof by induction, suitable for Grade 12 students.
Includes
Standards
Topics
Mathematical Proofs Worksheet
Name:
Date:
Score:
Read each question carefully and provide a complete and rigorous mathematical proof for each statement. Show all your steps and justify your reasoning.
1. Prove that if n is an odd integer, then n² is also an odd integer.
2. Prove by contradiction that the sum of a rational number and an irrational number is irrational.
3. Use mathematical induction to prove that for all positive integers n, the sum of the first n odd numbers is n². That is, 1 + 3 + 5 + ... + (2n - 1) = n².
Determine whether the following statements are true or false. Justify your answer with a brief explanation or counterexample.
4. The product of two irrational numbers is always an irrational number.
True
False
5. A proof that assumes the conclusion is false and then shows that this assumption leads to a contradiction is called a proof by .
6. The two main parts of a proof by mathematical induction are the base case and the step.
7. Prove that for any integers a, b, and c, if a | b and b | c, then a | c. (where 'a | b' means 'a divides b')