Mathematics 327  Sample Exam #1  3 October 2000

Problem 1. Suppose a is rational and b is irrational. Prove that a + b and ab are both irrational.

Problem 2. Suppose A ∩ B = A. Determine A ∪ B =?

Problem 3. Determine whether or not each of the binary relations R is reflexive symmetric, antisymmetric, or transitive:

a) A = {1, 2}, R = {(1, 2)}.

b) A = {1, 2, 3, 4}, R = {(1, 1), (1, 2), (2, 1), (3, 4)}.

c) A = ℤ, (a, b) ∈ R if and only if ab ≥ 0.

d) A = ℝ, (a, b) ∈ R if and only if a^{2} = b^{2}.

Problem 4. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9}. For a, b ∈ A define a ∼ b if ab is a perfect square (ie., it is a square of a natural number).

a) List ordered pairs of this relation.

b) For each a ∈ A find the ā = {x ∈ A | x ∼ a}.

c) Explain why ∼ is an equivalence relation on A.

Problem 5. Draw the Hesse diagram for the following partial order:

({{a}, {a, b}, {c}, {a, c}, {a, b, c}, {a, b, d}}, ⊆).

Problem 6. Let f : A → A be defined by f(x) = x^{2} + 2.

a) Let A = ℤ. Determine if f is injective and surjective.

b) Repeat part (a) for A = ℝ.

Problem 7. Let X = {a, b} and Y = {1, 2, 3}.

a) List all the functions from X to Y.

b) List all the functions from Y to X.

c) List all the injective functions from X to Y.

d) List all the surjective functions from X to Y.

Problem 8. Let S = {1, 2, 3, 4, 5}, and let f, g, h : S → S be the function defined by

f = {(1, 2), (2, 1), (3, 4), (4, 5), (5, 3)},
g = {(1, 3), (2, 5), (3, 1), (4, 2), (5, 4)},
h = {(1, 2), (2, 2), (3, 4), (4, 3), (5, 1)}.

a) Find f ∘ g and g ∘ f. b) Find f^{−1}, g^{−1}, and h^{−1} (if they exist).

c) Show that (f ∘ g)^{−1} = g^{−1} ∘ f^{−1} ≠ f^{−1} ∘ g^{−1}.

Problem 9.

a) Find the one-to-one correspondence between the intervals (1, ∞) and (3, ∞).

b) Find the one-to-one correspondence between the intervals (0, 1) and (a, b).

Problem 10.

a) Write the number 1001 in bases b = 2, 8.

b) Suppose that in base 12 we use A to denote 10 and B to denote 11. What is the number 1BBA?