Book of Proof - Amazon S3

214 Functions
If you continue your mathematical studies, you are likely to encounter
the following result in the future. For now, you are asked to prove it in
the exercises.
Theorem 12.4 Suppose f : A → B is a function. Let W, X ⊆ A, and Y , Z ⊆ B.
Then:
1. f (W ∩ X) ⊆ f (W) ∩ f (X)
2. f (W ∪ X) = f (W) ∪ f (X)
3. f −1 (Y ∩ Z) = f −1 (Y ) ∩ f −1 (Z)
4. f −1 (Y ∪ Z) = f −1 (Y ) ∪ f −1 (Z)
5. X ⊆ f −1 (f (X))
6. Y ⊆ f (f −1 (Y ))
Exercises for Section 12.6
1. Consider the function f : R → R defined as f (x) = x 2 + 3. Find f ([−3,5]) and
f −1 ([12,19]).
2. Consider the function f : { 1,2,3,4,5,6,7 } → { 0,1,2,3,4,5,6,7,8,9 } given as
f = { (1,3),(2,8),(3,3),(4,1),(5,2),(6,4),(7,6) } .
Find: f ({ 1,2,3 }) , f ({ 4,5,6,7 }) , f (), f −1({ 0,5,9 }) and f −1({ 0,3,5,9 }) .
3. This problem concerns functions f : { 1,2,3,4,5,6,7 } → { 0,1,2,3,4 } . How many
such functions have the property that ∣ ∣ f
−1 ({ 3 })∣ ∣ = 3?
4. This problem concerns functions f : { 1,2,3,4,5,6,7,8 } → { 0,1,2,3,4,5,6 } . How
many such functions have the property that ∣ ∣ f
−1 ({ 2 })∣ ∣ = 4?
5. Consider a function f : A → B and a subset X ⊆ A. We observed in Section 12.6
that f −1 (f (X)) ≠ X in general. However X ⊆ f −1 (f (X)) is always true. Prove this.
6. Given a function f : A → B and a subset Y ⊆ B, is f (f −1 (Y )) = Y always true?
Prove or give a counterexample.
7. Given a function f : A → B and subsets W, X ⊆ A, prove f (W ∩ X) ⊆ f (W) ∩ f (X).
8. Given a function f : A → B and subsets W, X ⊆ A, then f (W ∩ X) = f (W) ∩ f (X) is
false in general. Produce a counterexample.
9. Given a function f : A → B and subsets W, X ⊆ A, prove f (W ∪ X) = f (W) ∪ f (X).
10. Given f : A → B and subsets Y , Z ⊆ B, prove f −1 (Y ∩ Z) = f −1 (Y ) ∩ f −1 (Z).
11. Given f : A → B and subsets Y , Z ⊆ B, prove f −1 (Y ∪ Z) = f −1 (Y ) ∪ f −1 (Z).
12. Consider a function f : A → B. Prove that f is invertible if and only if X =
f −1 (f (X)) for every X ⊆ A.
13. Let f : A → B be a function, and X ⊆ A. Prove or disprove: f ( f −1 (f (X)) ) = f (X).
14. Letf : A → B be a function, and Y ⊆ B. Prove or disprove: f −1( f (f −1 (Y )) ) = f −1 (Y ).