Introduction to Mathematical Proofs

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Show your work for each problem.

1.
What is a mathematical proof?
1. a)A guess about whether something is true
2. b)A formula to memorize
3. c)A single example that works
4. d)A logical argument showing why a statement must be true
2.
What is the difference between an axiom and a theorem?
1. a)They are the same thing
2. b)Axioms are false; theorems are true
3. c)A theorem is accepted without proof; an axiom is proven
4. d)An axiom is accepted without proof; a theorem is proven
3.
If $n$ is an even number, we can write $n = 2k$ where $k$ is an integer. If $n = 12$ , what is $k$ ?
4.
If $n$ is an odd number, we can write $n = 2k + 1$ . If $n = 15$ , what is $k$ ?
5.
In a proof by contradiction, what do we do first?
1. a)Prove the statement directly
2. b)Give several examples
3. c)State the conclusion first
4. d)Assume the opposite of what we want to prove
6.
Why is showing $2 + 4 = 6$ NOT a proof that 'the sum of two even numbers is even'?
1. a)It only proves one specific case, not all cases
2. b)We need to use subtraction instead
3. c)The calculation is wrong
4. d)6 is not even
7.
Let $a = 2m$ and $b = 2n$ be even numbers. Complete: $a + b = 2m + 2n = 2(\square)$ . What goes in the box?
8.
Prove that the product of two odd numbers is odd. Let $a = 2m + 1$ and $b = 2n + 1$ .
9.
Which statement correctly describes circular reasoning in a proof?
1. a)Ending with QED
2. b)Starting with a definition
3. c)Using too many steps
4. d)Using the conclusion as part of the proof
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.
Prove: The sum of an even number and an odd number is odd.
12.
The statement 'All prime numbers are odd' seems true for 3, 5, 7, 11, 13... What disproves this statement?
1. a)1 is not prime
2. b)15 has many factors
3. c)9 is odd but not prime
4. d)2 is prime but even
13.
Prove by contradiction: There is no largest integer.
14.
Why must $a$ be even when $a^2$ is even? Suppose $a$ were odd, $a = 2k + 1$ . Then $a^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$ . What is the remainder when $a^2$ is divided by 2?
15.
If ' $P$ implies $Q$ ' is true, which of the following must also be true?
1. a)'Not $Q$ implies not $P$ ' (contrapositive)
2. b)' $Q$ implies $P$ ' (converse)
3. c)'Not $P$ implies not $Q$ ' (inverse)
4. d)All of the above
16.
Prove: If $n^2$ is odd, then $n$ is odd.