Download Practice Midterm 1

Math 300 - Introduction to Mathematical reasoning
Summer 2013
Practice Midterm 1
1. Suppose you have two statements
P (x, y) :
Q(x, y) :
x and y are two integers such that x + y is even.
x and y are two integers such that x and y are odd.
(a) Only one of these statements implies the other statement. Does P =⇒ Q or
does Q =⇒ P ? (No proof required.)
(b) Give a concrete example to show that reverse implication does not hold (No Proof
required).
2. Let A and B be two sets. Prove that A ∪ B = A ∪ (B − A).
3. Let f : X → Y and g : Y → X be two functions such that the composition g ◦ f is the
identity function (i.e. if x ∈ X, then (g ◦ f )(x) = x). Prove the following :
(a) f is injective.
(b) g is surjective.
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4. In this question, we shall prove that 3 is not a rational number. Please use the hints
provided (and NOT another technique).
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(a) In class, we have used
√ the fact that f (x) = x is an increasing function. Use this
to show that 3/2 < 3 < 2.
√
√
(b) Suppose that 3 was a positive rational number. Let us say that 3 is equal to
c/d, where c and d are natural numbers. Use part 4(a) to show that 3d − c < c.
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(c) Use the well-ordering principle to conclude that 3 is not a rational number.
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Lecture 2:
2-1
Math 300 - Introduction to Mathematical reasoning
Summer 2013
Practice Midterm 2
1. Negate the following statement without the using the word “NOT”
“The sequence {an } is bounded.”
2. Use induction to prove the following formula
12 + 22 + 32 + · · · + n2 =
n(n + 1)(2n + 1)
.
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3. Let P , Q and R be three logical statements. Two of the following three statements are
logically equivalent to each other.
(P =⇒ Q) =⇒ R
P =⇒ (Q =⇒ R)
(P =⇒ R) ∨ (P =⇒ ¬Q)
(a) Identify the two logically equivalent statements. Prove that they are logically
equivalent.
(b) Why is the remaining statement NOT logically equivalent to the two logically
equivalent statements in Part 4(a) ? (You must provide a concrete example where
this statement is not logically equivalent to the two other statements.)
4. Give a concrete function f : Z → N that is injective but NOT surjective. Specify a
natural number which is not contained in the image of f . (You do not have to provide
proofs. However, you must define your function f properly.)