# 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:
• $\forall x \exists y P(x, y)$
• $\exists x \forall y P(x, y)$
• $\forall x \exists y \lnot P(x, y)$
• $\exists y \forall x P(x, y)$
• $\exists x P(x, x)$
• $\forall x \lnot P(x, x)$