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

262 Chapter 5. Set Theory
Final Note. The definition of an ordered pair in Preview Activity 2 may have
seemed like a lengthy definition, but in some areas of mathematics, an even more
formal and precise definition of “ordered pair” is needed. This definition is explored
in Exercise (10).
Exercises for Section 5.4
? 1. Let A Df1; 2g, B Dfa; b; c; dg, andC Df1; a; bg. Use the roster method
to list all of the elements of each of the following sets:
(a) A B
(e) A .B \ C/
(b) B A
(f) .A B/ \ .A C/
(c) A C
(g) A ;
(d) A 2 (h) B f2g
2. Sketch a graph of each of the following Cartesian products in the Cartesian
plane.
(a) Œ0; 2 Œ1; 3
(b) .0; 2/ .1; 3
(c) Œ2; 3 f1g
(d) f1gŒ2; 3
(e) R .2; 4/
(f) .2; 4/ R
(g) R f 1g
(h) f 1gŒ1; C1/
? 3. Prove Theorem 5.25,Part(1): A .B \ C/ D .A B/ \ .A C/.
? 4. Prove Theorem 5.25,Part(4): .A [ B/ C D .A C/[ .B C/.
5. Prove Theorem 5.25,Part(5): A .B C/ D .A B/ .A C/.
6. Prove Theorem 5.25,Part(7): If T A, thenT B A B.
7. Let A Df1g;B Df2g; and C Df3g.
(a) Explain why A B ¤ B A.
(b) Explain why .A B/ C ¤ A .B C/.
8. Let A and B be nonempty sets. Prove that A B D B A if and only if
A D B.