Direct Proof

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Show your work for each problem.

1.
What is the first step in a direct proof?
1. a)Use proof by contradiction
2. b)Find a counterexample
3. c)Assume the hypothesis is true
4. d)Prove the conclusion directly
2.
According to the definition, an even integer can be written as:
1. a) for some integer
2. b) for some integer
3. c) for some integer
4. d) for some integer
3.
If for some integer , is even or odd? Type 'even' or 'odd'.
4.
What does QED stand for?
1. a)Quite easily done
2. b)Question every detail
3. c)Quod erat demonstrandum
4. d)Quickly end discussion
5.
If and , what is in terms of and ? Write your answer in the form .
6.
Why is checking examples not the same as a proof?
1. a)Examples take too long
2. b)Examples don't cover all infinite cases
3. c)Examples are always wrong
4. d)Examples are harder than proofs
7.
Prove: If is even, then is even.
8.
If (odd), what is when expanded? Write in the form .
9.
Prove: The product of two odd numbers is odd.
10.
What is circular reasoning in a proof?
1. a)Starting over multiple times
2. b)Going in a circle while writing
3. c)Using too many steps
4. d)Assuming what you're trying to prove
11.
In the factorization , how many consecutive integers are multiplied together?
12.
Prove: If is even and is odd, then is odd.
13.
To prove is divisible by 3, we use the fact that among any 3 consecutive integers:
1. a)None are divisible by 3
2. b)All are divisible by 3
3. c)At least two are divisible by 3
4. d)Exactly one is divisible by 3
14.
Prove: If is an integer, then is even.
15.
If we want to prove ' divides ' using direct proof, we show for some integer . What value of shows that 3 divides 12?
16.
Prove: The sum of an even number and its square is even.
17.
Which statement best describes an axiom?
1. a)A statement that needs verification
2. b)A statement that has been proven true
3. c)A statement that is always false
4. d)A statement accepted as true without proof