Abstract Algebra: Functions and Mappings

Determine whether the following mappings represent a function: Inputs: \{2, 4, 5\}, Outputs: \{-1, 0, 4\}.

Determine whether the following mappings represent a function: Inputs: {-4, 0, 8}, Outputs: {1, 2, 5, 7}.

Determine whether the following mappings represent a function: Inputs: {-3, -2, 0, 4}, Outputs: {-7, -5, -3}.

Describe a bijection $heta$ from the set mathbb{Z} of integers to its proper subset $E$, such that $E$ is the set of multiples of two of each integer. Show that the function $heta(x) = 2x$ is bijective.

Describe an injection from a set $X = \{x_1, x_2, \ldots, x_n\}$ with $n$ elements to the set $\mathbb{Z}$ of integers, where $\theta(x_i) = i$.

Let $\theta: \mathbb{R} \to \mathbb{N}_0$ be a mapping given by $\theta(x) = |x|$. Is $\theta$ injective? Explain.

List the elements of the complete inverse image $\theta^{-1}(5)$ under the mapping $\theta(x) = |x|$.

Prove that the function $f(x) = 3x - 2$ is injective.

Prove that the function $f(x) = x^2$ is not injective.

Prove that the function $f(x) = 5x + 2$ is surjective from the reals to the reals.

Determine if the function $f(x) = 5x + 2$ is surjective from the integers to the integers.

Find the inverse of the function $f(x) = 5x + 2$.