Mathematical Proofs Worksheet

Mathematical Proofs Worksheet

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.

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')