Introduction to Mathematical Proofs

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
Answer: A logical argument showing why a statement must be true
A mathematical proof is a logical argument that demonstrates why a statement must be true for all cases, using definitions, axioms, and previously proven theorems.
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
Answer: An axiom is accepted without proof; a theorem is proven
Axioms are statements we accept as true without proof - they are our starting points. Theorems are statements that have been proven true using axioms and other theorems.
3.
If is an even number, we can write where is an integer. If , what is ?
Answer: 6
Since , we solve for : . This confirms 12 is even because .
4.
If is an odd number, we can write . If , what is ?
Answer: 7
Since , we solve: , so . Check: .
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
Answer: Assume the opposite of what we want to prove
In proof by contradiction, we assume the opposite of what we want to prove is true. Then we show this leads to a logical impossibility (contradiction), proving our original statement must be true.
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
Answer: It only proves one specific case, not all cases
There are infinitely many pairs of even numbers. Testing only proves that ONE pair works. A proof must show ALL pairs work, which requires using variables like .
7.
If and are even numbers, what is in terms of and ? Write in the form
Answer: 2(m+n)
. Since is an integer, is even. This proves the sum of two even numbers is always even!
8.
Prove that the product of two odd numbers is odd. Let and .
Answer: 2(2mn+m+n)+1
1. Write out expanded 4mn+2m+2n+1
2. Factor out 2 from the first three terms 2(2mn+m+n)+1
3. Is in the form ? yes
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
Answer: Using the conclusion as part of the proof
Circular reasoning occurs when you assume what you're trying to prove. For example, saying 'X is true because X is true' doesn't actually prove anything - it just goes in circles.
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?
Answer: 2
If both and are even, they both have 2 as a factor. This contradicts our assumption that they share no common factors. This contradiction proves cannot be written as .
11.
Prove: The sum of an even number and an odd number is odd.
Answer: 2(m+n)+1
1. Let the even number be . Write the odd number in the form 2n+1
2. Calculate 2m+2n+1
3. Rewrite by factoring: 2(m+n)+1
4. Is odd? (yes/no) yes
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
Answer: 2 is prime but even
2 is prime (only divisible by 1 and itself) but it is even. One counterexample is enough to disprove a universal statement. This shows why we need proofs, not just examples!
13.
Prove by contradiction: There is no largest integer.
Answer: N+1
1. Assume the opposite: there IS a largest integer. Call it . Is also an integer? yes
2. Is ? yes
3. We found an integer larger than . What is it? N+1
4. This contradicts that is the largest. Therefore, our assumption was... (true/false) false
14.
If is even, then must be even. Why? Because if were odd (), then , which is... (even/odd)?
Answer: odd
is in the form , which is odd. So if were odd, would be odd. Therefore, if is even, cannot be odd - it must be even!
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
Answer: 'Not implies not ' (contrapositive)
The contrapositive ('Not implies not ') is always logically equivalent to the original statement. The converse and inverse are NOT guaranteed to be true. Example: 'If it rains, the ground is wet' is true, but 'If the ground is wet, it rained' is not always true (sprinklers!).
16.
Prove: If is odd, then is odd.
Answer: n is odd
1. We'll prove the contrapositive. If is even, then is... (even/odd)? even
2. If (even), what is ? 4k^2
3. Write in the form 2(2k^2)
4. Since is even, we proved: if is even then is even. By contrapositive, if is odd, then is... odd