Page 240 - HOW TO PROVE IT: A Structured Approach, Second Edition

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5

Functions

5.1. Functions

Suppose P is the set of all people, and let H ={(p, n) ∈ P × N | the person
p has n children}. Then H is a relation from P to N, and it has the following
important property. For every p ∈ P, there is exactly one n ∈ N such that
(p, n) ∈ H. Mathematicians express this by saying that H is a function from
P to N.

Definition 5.1.1. Suppose F is a relation from A to B. Then F is called a
function from A to B if for every a ∈ A there is exactly one b ∈ B such that
(a, b) ∈ F. In other words, to say that F is a function from A to B means:

∀a ∈ A∃!b ∈ B((a, b) ∈ F).

To indicate that F is a function from A to B, we will write F : A → B.

Example 5.1.2.

1. Let A ={1, 2, 3}, B ={4, 5, 6}, and F ={(1, 5), (2, 4), (3, 5)}.Is F a
function from A to B?
2. Let A ={1, 2, 3}, B ={4, 5, 6}, and G ={(1, 5), (2, 4), (1, 6)}.Is G a
function from A to B?
3. Let C be the set of all cities, N the set of all countries, and let L ={(c, n) ∈
C × N | the city c is in the country n}.Is L a function from C to N?
4. Let P be the set of all people, and let C ={(p, q) ∈ P × P | the person p
is a parent of the person q}.Is C a function from P to P?
5. Let P be the set of all people, and let D ={(p, x) ∈ P × P (P) | x = the
set of all children of p}.Is D a function from P to P (P)?

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