Homework 3

Due Wednesday, 3/14:

  1. If X is a set, the set X^2 = X \times X is the set of all ordered pairs of elements of X. Define a function f : \mathbb{R} \to \mathbb{R}^2 by f(x) = (\mathrm{cos}(x), \mathrm{sin}(x)).
    • Is (0, 0) an element of the range of f?
    • State four elements of the range of f.
    • Determine if f is an injection, a surjection, both (a bijection), or neither. Prove your answers.
  2. Let g : \mathbb{R}^2 \to \mathbb{R} be defined by g(x, y) = x. Determine if g is an injection, a surjection, both, or neither. Prove your answers.
  3. Let A = { 1, 2, 3, …, n } and B = { a, b, c }. If n = 1, how many functions are there from A to B (ie, with domain A and codomain B? If n = 2, how many are there? Find a formula for the number of functions from a set A of cardinality n to the set B (which has cardinality 3).
  4. Let \mathcal{F} be the set of functions with domain { 1, 2, 3 } and codomain { 0, 1 }. That is, f \in \mathcal{F} if f : \{ 1, 2, 3 \} \to \{ 0, 1 \}. Find a bijection from \mathcal{F} to \mathcal{P}(\{1, 2, 3\}). That is, find a bijection whose domain is the set of functions from {1, 2, 3} to {0, 1} and whose codomain is the set of subsets of {1, 2, 3}.
  5. Suppose our universe is \mathbb{Z} and P(x, y) is defined as x + 1 = y. Determine which of the following statements are true and explain why:
    • \forall x \exists y P(x, y)
    • \exists x \forall y P(x, y)
    • \forall x \exists y \lnot P(x, y)
    • \exists y \forall x P(x, y)
    • \exists x P(x, x)
    • \forall x \lnot P(x, x)