Introduction to Mathematical Proofs

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Show your work for each problem.

1.
What is a mathematical proof?
1. a)A logical argument showing why a statement must be true
2. b)A formula to memorize
3. c)A guess about whether something is true
4. d)A single example that works
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)An axiom is accepted without proof; a theorem is proven
4. d)A theorem is accepted without proof; an axiom is proven
3.
If is an even number, we can write where is an integer. If , what is ?
4.
If is an odd number, we can write . If , what is ?
5.
In a proof by contradiction, what do we do first?
1. a)Assume the opposite of what we want to prove
2. b)Prove the statement directly
3. c)Give several examples
4. d)State the conclusion first
6.
Why is showing NOT a proof that 'the sum of two even numbers is even'?
1. a)The calculation is wrong
2. b)We need to use subtraction instead
3. c)It only proves one specific case, not all cases
4. d)6 is not even
7.
If and are even numbers, what is in terms of and ? Write in the form
8.
Prove that the product of two odd numbers is odd. Let and .
9.
Which statement correctly describes circular reasoning in a proof?
1. a)Ending with QED
2. b)Using too many steps
3. c)Using the conclusion as part of the proof
4. d)Starting with a definition
10.
In the proof that is irrational, we assumed with no common factors. We showed both and 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)9 is odd but not prime
2. b)1 is not prime
3. c)2 is prime but even
4. d)15 has many factors
13.
Prove by contradiction: There is no largest integer.
14.
If is even, then must be even. Why? Because if were odd (), then , which is... (even/odd)?
15.
If ' implies ' is true, which of the following must also be true?
1. a)'Not implies not ' (contrapositive)
2. b)' implies ' (converse)
3. c)'Not implies not ' (inverse)
4. d)All of the above
16.
Prove: If is odd, then is odd.