1.1. Sets and Functions 7

Cartesian Product. If A and B are sets, then their Cartesian product A × B

is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. Similarly, the Cartesian

product of n sets A1,A2,...,An is the set A1 ×A2 ×· · ·×An of all ordered n-tuples

(a1,a2,...,an) with ai ∈ Ai for i = 1,...,n.

If f : A → B is a function from a set A to a set B, then the graph of f is the

subset of A × B defined by {(a, b) ∈ A × B : b = f(a)}.

Exercise Set 1.1

1. If a, b ∈ R and a b, give a description in set theory notation for each of the

intervals (a, b), [a, b], [a, b), and (a, b] (see Example 1.1.1).

2. If A, B, and C are sets, prove that

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

3. If A and B are two sets, then prove that A is the union of a disjoint pair of

sets, one of which is contained in B and one of which is disjoint from B.

4. What is the intersection of all the open intervals containing the closed interval

[0,1]? Justify your answer.

5. What is the intersection of all the closed intervals containing the open interval

(0,1)? Justify your answer.

6. What is the union of all of the closed intervals contained in the open interval

(0,1)? Justify your answer.

7. If A is a collection of subsets of a set X, formulate and prove a theorem like

Theorem 1.1.5 for the intersection and union of A.

8. Which of the following functions f : R → R are one-to-one and which ones are

onto. Justify your answer.

(a) f(x) =

x2;

(b) f(x) =

x3;

(c) f(x) =

ex.

9. Prove part (b) of Theorem 1.1.6.

10. Prove part (c) of Theorem 1.1.6.

11. Prove part (a) of Theorem 1.1.7.

12. Prove part (b) of Theorem 1.1.7.

13. Give an example of a function f : A → B and subsets F ⊂ E of A for which

f(E) \ f(F) = f(E \ F).

14. Prove that equality holds in parts (b) and (c) of Theorem 1.1.7 if the function

f is one-to-one.

15. Prove that if f : A → B is a function which is one-to-one and onto, then f

has an inverse function – that is, there is a function g : B → A such that

g(f(x)) = x for all x ∈ A and f(g(y)) = y for all y ∈ B.