Relations and Functions

Consider the relation .

1. Is $R$ a function?
2. Find $R^{-1}$.
3. Is $R^{-1}$ a function? Why or why not?

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. Is $f^{-1}$ a function? Explain.

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 $f(n) = \begin{cases}n/2 & \text{ if } n \text{ is even} \\ (n+1)/2 & \text{ if } n \text{ is odd} . \end{cases}$

Determine whether $f:\Z \times \Z \to \Z$ is surjective if

1. $f(m,n) = m + n$
2. $f(m,n) = m^2 + n^2$
3. $f(m,n) = m$
4. $f(m,n) = m - n$

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.

Give an explicit formula for a function from the set of integers to the set of positive integers that is

1. injective but not surjective
2. surjective but not injective
3. injective and surjective
4. neither injective nor surjective

What can you deduce about the sets $X$ and $Y$ if you know …

1. there is an injective function $f:X \to Y$? Explain.

2. there is a surjective function $f:X \to Y$? Explain.

3. there is a bijective function $f:X \to Y$? Explain.

Suppose $f:X \to Y$ is a function. Which of the following are possible? Explain.

1. $f$ is injective but not surjective.
2. $f$ is surjective but not injective.
3. $|X| = |Y|$ and $f$ is injective but not surjective.
4. $|X| = |Y|$ and $f$ is surjective but not injective.
5. $|X| = |Y|$, $X$ and $Y$ are finite, and $f$ is injective but not surjective.
6. $|X| = |Y|$, $X$ and $Y$ are finite, and $f$ is surjective but not injective.

Find the floor and ceiling of each of the following real numbers:

1. 8.675309
2. $\pi$
3. $-2.1$
4. $7$

Let $f:Z_5 \to Z_5$ be defined by $f(n) = \left \lfloor \dfrac{n}{2}\right \rfloor$. Complete the table of values:

Is $f$ injective? surjective?

Are the floor or ceiling functions injective? Surjective? If so, prove it. If not, provide specific counterexamples to show why not.