Discrete Mathematics for Computer Science

226 CHAPTER 4 Functions

y

(-3, 0) 1 3, 0)

(0,-3) (-•'1)

Figure 4.3 Graph of a relation that is not a function.

we often represent such functions by a diagram such as that shown on Figure 4.4. The

lines joining an element on the left in Figure 4.4 with an element on the right represent

the association between elements of the domain and elements of the codomain that we

interpret as the rule for F. For example, we interpret the line between 0 and 5 as meaning

F(0) = 5.

F: {0, 1,2) {3, 5, 7}

1 15

2 -7

Figure 4.4 Representation of a function.

The elements of the domain and of the codomain can be listed in any order. Sometimes,

a picture of this sort makes functions defined on N easier to understand. This representation

can also be used for some "large" sets.

4.1.5 Equality of Functions

Since functions are defined as subsets of a product of two sets-that is, as sets of ordered

pairs-two functions are equal when they are equal as sets.

Definition 3. Let F, G : X -+ Y be two functions. The functions F and G are equal if

and only if they contain the same ordered pairs.

Example 13. Let SqrN be the function from N to N defined by the rule SqrN(n) = n^2.

Let SqrR be the function from R to R defined by the rule SqrR(r) = r^2 .Then, SqrNq and

SqrR are not the same function, since (1.1, 1.21) r SqrR but (1.1, 1.21) 0 Sqri.