Bridge to Abstract Mathematics: Mathematical Proof and Structures

26 SETS Chapter 1

-..

(A U B) n C.is the

crosshatched region

Figure 1.5 A more complicated Venn

diagram representation.

operation of set theory. For example, the set (A u B) n C may be repre-

sented by the -- doubly shaded region in the Venn diagram, Figure 1.5.

We will see in the next article how Venn diagrams can be useful for sug-

gesting theorems of set theory; later we will encounter other interesting

uses. In Article 1.5 they prove helpful for solving certain counting problems;

in Article 3.5 we use them as an aid in analyzing the logical validity of

certain kinds of arguments.

Exercises

In Exercises 1 through 4, let U = {1,2, 3,... ,9, 101, A = (1, 7,9), B =

(3, 5,6,9, 101, and C = (2,4, 8,9).

1. Calculate:
   (a) B u B
   (c) A u A'
   (e) (AuB)nA
   (9) B'
   (i) A - A
   (k) A A A


2. Calculate:
   (a) A n C
   (c) A n C'
   (e) C - B
   fs) (AuB)uC
   (i) A u (B u C)
   *(k) (A u B) n C
   (m) (A u B) u (A u B')
   (0) A u (C n A')
   (9) (B U C)'


3. Calculate:
   (a) (A n B n C)'

(b) C n C

(d) B n B'

(f) (B n C) u C

, fh) B"

(1) B - B'

*(I) CAC'

(A n C)'

A' u C'

CnB'

A n (B n C)

(A n B) n C

(A u C) n (B u C)

(A n C) u (B n C)

(A n C) u (A n C')

B' u C'

(b) A' u B' u C'