Direct Proof

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
Answer: Assume the hypothesis is true
In a direct proof, we start by assuming the hypothesis (the 'if' part) is true, then use logical steps to reach the conclusion.
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
Answer: for some integer
An even integer is any integer that is divisible by 2, so it can be written as where is any integer. For example: , .
3.
If for some integer , is even or odd? Type 'even' or 'odd'.
Answer: odd
The expression represents an odd number because is even (a multiple of 2) and adding 1 makes it 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
Answer: Quod erat demonstrandum
QED stands for 'quod erat demonstrandum', which is Latin for 'which was to be demonstrated'. It marks the successful completion of a proof.
5.
If and , what is in terms of and ? Write your answer in the form .
Answer: 2(m + n)
. This factored form shows that the sum is even because it's 2 times an integer.
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
Answer: Examples don't cover all infinite cases
There are infinitely many integers, so we can never check all cases by example. A proof using variables and definitions covers all cases at once.
7.
Prove: If is even, then is even.
Answer: n^2 = 4k^2 = 2(2k^2)
1. Write using the definition of even (in terms of some integer ): n = 2k
2. Square both sides to find : n^2 = 4k^2
3. Factor out 2 to show is even: n^2 = 2(2k^2)
8.
If (odd), what is when expanded? Write in the form .
Answer: 4k^2 + 4k + 1
. This can be rewritten as , showing it's odd.
9.
Prove: The product of two odd numbers is odd.
Answer: 2(2jk + j + k) + 1
1. Let the two odd numbers be and . Find : (2j + 1)(2k + 1)
2. Expand the product using FOIL: 4jk + 2j + 2k + 1
3. Factor to show it's of the form (odd): 2(2jk + j + k) + 1
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
Answer: Assuming what you're trying to prove
Circular reasoning occurs when you use the conclusion (what you're trying to prove) as part of your reasoning. This makes the proof invalid because you haven't actually proven anything.
11.
In the factorization , how many consecutive integers are multiplied together?
Answer: 3
The expression factors into three consecutive integers: , , and . This is key to proving that is divisible by 3.
12.
Prove: If is even and is odd, then is odd.
Answer: 2(m + k) + 1
1. Write as an even number and as an odd number: a = 2m, b = 2k + 1
2. Add : 2m + 2k + 1
3. Factor to show it's odd (form ): 2(m + k) + 1
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
Answer: Exactly one is divisible by 3
Every integer has remainder 0, 1, or 2 when divided by 3. In three consecutive integers, we cycle through all three remainders, so exactly one has remainder 0 (divisible by 3).
14.
Prove: If is an integer, then is even.
Answer: n(n + 1)
1. Factor : n(n + 1)
2. What can we say about and ? consecutive integers
3. Among two consecutive integers, how many are even? 1
4. Therefore, the product is always: even
15.
If we want to prove ' divides ' using direct proof, we show for some integer . What value of shows that 3 divides 12?
Answer: 4
Since , we have . This demonstrates that 3 divides 12 because 12 can be expressed as 3 times an integer (4).
16.
Prove: The sum of an even number and its square is even.
Answer: 2(k + 2k^2)
1. Let the even number be . Write : 2k + (2k)^2
2. Simplify : 4k^2
3. Write in factored form : 2(k + 2k^2)
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
Answer: A statement accepted as true without proof
An axiom is a fundamental statement that is accepted as true without proof. Axioms serve as the foundation from which theorems are derived. Examples include 'a = a' (reflexive property) and 'if a = b, then b = a' (symmetric property).