HOMEWORK 1 FOR MATH 281
INTRODUCTION TO MATHEMATICAL REASONING
Fall 2008
Exercise 1. Choose ONE of the statements below and carefully prove it.
1.
(A ∩ B)c = Ac ∪ B c
2.
(A ∪ B)c = Ac ∩ B c
3.
(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
4.
Acc = A
Exercise 2. What is the cartesian product A × φ? Explain.
Exercise 3. Decide which of the relations below is an equivalence relation.
Explain your answer.
a∼b
if:
1. a is a blood relative to b.
2. a − b is divisible by 3.
3. a lives in the same town as b.
4. a and b have at some point lived in the same town.
5. a ≤ b.
6. a and b are siblilings.
Remember in class we pointed out that a relation on A is nothing but a
subset R ⊆ A × A
Exercise 4. Is the following subset an equivalence relation? If it is, describe
the quotient set.
S ⊆ R × R = {(x, y) : x ∈ Z, y ∈ Z} ∪ {(x, x) : x ∈ R}
Exercise 5. View the following function as a composition of simple functions:
p
log(sin2 (x))
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Exercise 6. What can we say about the composite function g ◦ f if:
• both f, g are injective?
• f is injective and g is surjective?
• g is injective and f is surjective?
• both f, g are surjective?
Explain!
Exercise 7. Show that g ◦ f bijective ⇒ f injective and g surjective.
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