Functions

Consider the function $f:\{1,2,3,4,5\} \to \{1,2,3,4\}$ given by the table below:

1. Is $f$ injective? Explain.

2. Is $f$ surjective? Explain.

3. Write the function using two-line notation.

For each function given below, determine whether or not the function is injective and whether or not the function is surjective.

1. $f:\N \to \N$ given by $f(n) = n+4$.
2. $f:\Z \to \Z$ given by $f(n) = n+4$.
3. $f:\Z \to \Z$ given by $f(n) = 5n - 8$.
4. $f:\Z \to \Z$ given by $f(n) = \begin{cases}n/2 & \text{ if } n \text{ is even} \\ (n+1)/2 & \text{ if } n \text{ is odd} . \end{cases}$

Let $A = \{1,2,3,\ldots,10\}$. Consider the function $f:\pow(A) \to \N$ given by $f(B) = |B|$. That is, $f$ takes a subset of $A$ as an input and outputs the cardinality of that set.

1. Is $f$ injective? Prove your answer.

2. Is $f$ surjective? Prove your answer.

3. Find $f\inv(1)$.

4. Find $f\inv(0)$ .

5. Find $f\inv(12)$.

Let $f:X \to Y$ be some function. Suppose $3 \in Y$. What can you say about $f\inv(3)$ if you know,

1. $f$ is injective? Explain.

2. $f$ is surjective? Explain.

3. $f$ is bijective? Explain.

Find a set $X$ and a function $f:X \to \N$ so that $f\inv(0) \cup f\inv(1) = X$.

Consider the function $f:\Z \to \Z$ given by $f(n) = \begin{cases}n+1 & \text{ if }n\text{ is even} \\ n-3 & \text{ if }n\text{ is odd} . \end{cases}$

1. Is $f$ injective? Prove your answer.

2. Is $f$ surjective? Prove your answer.

At the end of the semester a teacher assigns letter grades to each of her students. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bijective, or neither?

Seven players are playing 5-card stud. Each player initially receives 5 cards from a deck of 52. Is this a function? If so, what sets make up the domain and codomain, and is the function injective, surjective, bijective, or neither?

Consider the function $f:\N \to \N$ that gives the number of handshakes that take place in a room of $n$ people assuming everyone shakes hands with everyone else. Give a recursive definition for this function.

Let $f:X \to Y$ be a function and $A \subseteq X$ be a finite subset of the domain. What can you say about the relationship between $\card{A}$ and $\card{f(A)}$? Consider both the general case and what happens when you know $f$ is injective, surjective, or bijective.

Let $f:X \to Y$ be a function and $B \subseteq Y$ be a finite subset of the codomain. What can you say about the relationship between $\card{B}$ and $\card{f\inv(B)}$? Consider both the general case and what happens when you know $f$ is injective, surjective, or bijective.