Mathematics 327  Exam #1  9 October 2000

MATHEMATICS 327

NAME:

1. 2. 3. 4. 5.

BONUS PROBLEM. TOTAL

Mathematics 327  Exam #1  9 October 2000

Problem 1. Let A, B, C be sets. Show that, in general,

(A \ B) \ C ≠ A \ (B \ C).

Hint: Give any example when this equality is violated.

Problem 2. Determine whether or not each of the binary relations R is reflexive, symmetric, antisymmetric, or transitive:

a) A = {1, 2, 3, 4}, R = {(1, 1), (1, 2), (2, 1), (3, 4), (4, 3)}. reflexive: symmetric: antisymmetric: transitive:

b) A = ℝ, (a, b) ∈ R if and only if a − b ≤ 3. reflexive: symmetric: antisymmetric: transitive: c) A = ℤ, (a, b) ∈ R if and only if a + b = 10. reflexive: symmetric: antisymmetric: transitive:

d) A = ℕ, (a, b) ∈ R if and only if \frac{a}{b} ∈ ℕ. reflexive: symmetric: antisymmetric: transitive:

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

a) Find f ∘ g and g ∘ f.

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

c) Find f^{2}, f^{3}, f^{4}. What is f^{n}? (Here f^{2} = f ∘ f, f^{3} = f ∘ f ∘ f, ...)

Problem 4. In each case determine if the function is injective (1-1) and/or surjective (onto):

a) f : ℕ → ℕ, f(n) = 3n:

b) f : ℝ → ℝ, f(x) = 3x:

c) f : ℝ^{2} → ℝ, f(x, y) = x:

d) f : ℝ → ℝ, f(x) = x^{4} − x^{2}:

Problem 5.

a) Write the number 10001 in base b = 2.

b) Let x = (10001)_{3} and y = (111)_{3}. Compute the sum x + y and the product x · y in base b = 3.

BONUS PROBLEM. Suppose m, n ∈ ℤ and n^{2} + 1 = 2m. Prove that m is a sum of two squares (i.e., m = p^{2} + q^{2} where p, q are some integers).