# Determine Total/Partial Functions and Injection/Surjection/Bijection

I understand the concept of determining Functions vs. Partial Functions, as well as what injection, bijection, and surjection are. However, some HW problems given to me have domains and ranges using real numbers (R) that I don't quite grasp.

1. Classify each of the following as a function, a partial function, or not a function. (If it’s both a function and a partial function, choose “function”—the more exact classification.)

a) f = {(1, a)} for domain space {1} and range space {a, b}.

My Answer: Function (all domains used exactly once)

b) f(x) = 1/x when both domain and range space are the Real numbers (R).

c) f(x) = sin(x) when domain space is the Natural numbers (N, the non-negative integers) and range space is R.

d) f(x) = x$^{½}$ when both domain and range space are R. (Recall that the square root of a positive number n is $\pm$n.)

e) f(x, y) = max(x, y) for domain space R $\times$ R and range space R.

f) f = {(<1, 1>, a), (<2, 1>, b), (<1, 2>, b)} for domain space {1, 2} $\times$ {1, 2} and range space {a, b}.

Domain(?): {<1,1>,<1,2>,<2,1>,<2,2>}

My Answer: Partial Function (Not all domains used; none used more than once)

g) f = {(<1, 1>, a), (<1, 1>, b), (<1, 2>, b)} for domain space {1} $\times$ {1, 2} and range space {a, b}.

Domain(?): {<1,1>,<1,2>}

My Answer: NOT a function (Some domains used more than once)

1. Classify as injection, surjection, bijection, or none. Give the most specific answer.

f(x) = x$^{2}$+1 when both domain and range space are positive R.

My Answer: Bijection(?) (Never any repeats in domain or range; each value in range used exactly once)

Your "partial function" is not a standard concept, so I had to figure out the meaning from clues in your post. Apparently it is:

The relation is not defined on every element of the domain, but where it is defined, it is defined uniquely (functionally).

Based on this, all the answers you gave are correct. Now for the ones you didn't give:

b) $f : \Bbb R \to \Bbb R : x \mapsto 1/x$ (this is the common short notation for defining domain, range, and formula for a function).

The questions you need to ask yourself are:

1. Is $f(x)$ defined for every $x \in \Bbb R$?
2. Does $f(x)$ have more than one value for some single value of $x$?

If 2. is true, then $f$ is not a function (or partial function). If 2. is false, and 1. is false, then $f$ is a partial function. If 2. is false, and 1. is true, then $f$ is a function.

In this case, the answers are

1. No. (What number can you not divide by?)
2. No. (Multiplicative inverses are unique.)

So it is a partial function.

c) $f : \Bbb N \to \Bbb R : x \mapsto \sin x$

1. Is $f(x)$ (i.e., $\sin x$) defined for every $x \in \Bbb N$? (It is defined for a lot of values outside of $\Bbb N$, but we are not concerned about that here. Here the question is just: are there any natural numbers for which it isn't defined?)
2. Does $\sin x$ ever have more than one value for the same value of $x$?

I'll leave it to you to figure out the answers.

d) $f : \Bbb R \to \Bbb R : x \mapsto x^{1/2}$

(This is somewhat poorly written, because by standard convention $x^{1/2}$ refers to the positive root only. However in this case the question clearly indicates that both roots are intended, so that is what you should assume.)

1. Is $x^{1/2}$ defined (as a real number) for every $x \in \Bbb R$?
2. Does $x^{1/2}$ ever have more than one value for the same value of $x$?

e) $f : \Bbb R \times \Bbb R \to \Bbb R : (x, y) \mapsto \max\{x,y\}$

1. Is $\max\{x,y\}$ defined for every $(x, y) \in \Bbb R \times \Bbb R$?
2. Does $\max\{x,y\}$ ever have more than one value for the same values of $x$ and $y$?