Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

6.6. Functions Acting on Sets 357
Theorem 6.36. Let f W S ! T be a function and let A be a subset of S and let C
be a subset of T .Then
1. A f 1 .f .A// 2. f f 1 .C / C
Proof. We will prove Part (1). The proof of Part (2) isExercise(7).
To prove Part (1), we will prove that for all a 2 S, if a 2 A, then
a 2 f
1 .f .A//. So let a 2 A. Then, by definition, f.a/ 2 f.A/. We know
that f.A/ T ,andsof
1 .f .A// S. Notice that
f 1 .f .A// Dfx 2 S j f.x/ 2 f.A/g :
Since f.a/ 2 f.A/, we use this to conclude that a 2 f
1 .f .A//. This proves
that if a 2 A, thena 2 f 1 .f .A//, and hence that A f 1 .f .A//.
Exercises 6.6
1. Let f W S ! T ,letA and B be subsets of S, andletC and D be subsets of
T .Forx 2 S and y 2 T , carefully explain what it means to say that
? (a) y 2 f.A\ B/
(b) y 2 f.A[ B/
(c) y 2 f.A/\ f.B/
? (d) y 2 f.A/[ f.B/
(e) x 2 f 1 .C \ D/
? (f) x 2 f 1 .C [ D/
(g) x 2 f 1 .C / \ f 1 .D/
? (h) x 2 f 1 .C / [ f 1 .D/
2. Let f W R ! R by f.x/ D 2x C 1. Let
A D Œ2; 5 B D Œ 1; 3 C D Œ 2; 3 D D Œ1; 4.
Find each of the following:
(a) f.A/
? (b) f 1 .f .A//
(c) f 1 .C /
? (d) f.f 1 .C //
? (e) f.A\ B/
? (f) f.A/\ f.B/
(g) f 1 .C \ D/
(h) f 1 .C / \ f 1 .D/