Equality of Functions

Section 7.2 Equality of Functions

Two functions are equal if they have the same domain and codomain and their values are the same for all elements of the domain.

Definition 7.2.1.

Let \(A\) and \(B\) be sets and \(f:A\to B\) and \(g:A\to B\) be functions. We say that \(f\) and \(g\) are equal and write \(f=g\) if \(f(a)=g(a)\) for all \(a\in A\text{.}\) If \(f\) and \(g\) are not equal, we write \(f\ne g\text{.}\)

In our definition of the equality of functions, we have assumed that the two functions have the same domain and codomain. Two functions that do not have the same domain and codomain are not equal.

In Checkpoint 7.2.2 fill in the blanks to

Checkpoint 7.2.2. Definition of equality of functions.

Complete the following:

Let f : A \(\rightarrow\) B and g : A \(\rightarrow\) B be functions. We say that the function f

select
in not equal to
is equal to
is similar to
is like

the function g and write f = g if f(c) = g(c)

select
for one element c of A
for one element c of B
for some elements c of A
for some elements c of B
for all elements c of A
for all elements c of B

Answer 1.

\(\text{is equal to}\)

Answer 2.

\(\text{for all elements c of A}\)

In the video in Figure 7.2.3 we recall the definition and show how one can determine whether two functions are equal.

Figure 7.2.3. Equality of functions

Example 7.2.4. Equal functions.

Let \(f:\Z_5\to\Z_5\) be given by \(f(a):=(a+1)\bmod 5\) and \(g:\Z_5\to\Z_5\) be given by \(g(a):=(a-4)\bmod 5\text{.}\) First, note that the domains of \(f\) and \(g\) are the same and the codomains of \(f\) and \(g\) are the same. We show that \(f=g\) by evaluating both \(f\) and \(g\) at each element of the common domain \(\Z_5=\{0,1,2,3,4\}\) and then comparing the function values for the same elements.

\(f(0)=(0+1)\bmod 5=1\)	\(g(0)=(0-4)\bmod 5=1\)
\(f(1)=(1+1)\bmod 5=2\)	\(g(1)=(1-4)\bmod 5=2\)
\(f(2)=(2+1)\bmod 5=3\)	\(g(2)=(2-4)\bmod 5=3\)
\(f(3)=(3+1)\bmod 5=4\)	\(g(3)=(3-4)\bmod 5=4\)
\(f(4)=(4+1)\bmod5=0\)	\(g(4)=(4-4)\bmod 5=0\)

Since \(f(a)=g(a)\) for all \(a\in\Z_5\text{,}\) we have \(f=g\text{.}\)

Problem 7.2.5. Are these functions equal ?

Decide whether the two functions

\begin{align*} f:\Z_3\to \Z_3 \amp \text{ given by } f(x)=(x^2+1)\fmod 3\\ g:\Z_3\to \Z_3 \amp \text{ given by } f(x)=(x-2)\fmod 3 \end{align*}

are equal.

Solution.

We evaluate \(f\) and \(g\) at elements of their domain \(\Z_3=\{0,1,2\}\text{.}\) If \(f(x)=g(x)\) for all \(x\in\Z_3\) then \(f=g\text{.}\)

\(f(0)=(0^2+1)\fmod 3 = 1\)	\(g(0)=(0-2) \fmod 3 = 1\)
\(f(1)=(1^2+1)\fmod 3 = 2\)	\(g(1)=(1-2) \fmod 3 = 2\)
\(f(2)=(2^2+1)\fmod 3 = 2\)	\(g(2)=(2-2) \fmod 3 = 0\)