Introduction to Mathematical Proofs

Practice Problems: Introduction to Mathematical Proofs

1. 1. What is a mathematical proof?
2. 2. What is the difference between an axiom and a theorem?
3. 3. If $n$ is an even number, we can write $n = 2k$ where $k$ is an integer. If $n = 12$, what is $k$?
4. 4. If $n$ is an odd number, we can write $n = 2k + 1$. If $n = 15$, what is $k$?
5. 5. In a proof by contradiction, what do we do first?
6. 6. Why is showing $2 + 4 = 6$ NOT a proof that 'the sum of two even numbers is even'?
7. 7. If $a = 2m$ and $b = 2n$ are even numbers, what is $a + b$ in terms of $m$ and $n$? Write in the form $2(...)$
8. 8. Prove that the product of two odd numbers is odd. Let $a = 2m + 1$ and $b = 2n + 1$.
9. 9. Which statement correctly describes circular reasoning in a proof?
10. 10. In the proof that $\sqrt{2}$ is irrational, we assumed $\sqrt{2} = \frac{a}{b}$ with no common factors. We showed both $a$ and $b$ must be even. What common factor do they share?
11. 11. Prove: The sum of an even number and an odd number is odd.
12. 12. The statement 'All prime numbers are odd' seems true for 3, 5, 7, 11, 13... What disproves this statement?
13. 13. Prove by contradiction: There is no largest integer.
14. 14. If $a^2$ is even, then $a$ must be even. Why? Because if $a$ were odd ($a = 2k+1$), then $a^2 = 4k^2 + 4k + 1 = 2(2k^2+2k) + 1$, which is... (even/odd)?
15. 15. If '$P$ implies $Q$' is true, which of the following must also be true?
16. 16. Prove: If $n^2$ is odd, then $n$ is odd.