Page 27 - Fundamentals of Probability and Statistics for Engineers

10 Fundamentals of Probability and Statistics for Engineers

B B
A A

(a) A ∪ B (b) A ∩ B
Figure 2.3 (a) Union and (b) intersection of sets A and B

The intersection or product of A and B, written as A B, or simply AB, is the
\
set of all elements that are common to A and B.
In terms of Venn diagrams, results of the above operations are shown in
Figures 2.3(a) and 2.3(b) as sets having shaded areas.
If AB ˆ; , sets A and B contain no common elements, and we call A and B
‡
disjoint. The symbol ‘ ’ shall be reserved to denote the union of two disjoint
sets when it is advantageous to do so.

Example 2.2. Let A be the set of all men and B consist of all men and women

over 18 years of age. Then the set A B consists of all men as well as all women
[
\
over 18 years of age. The elements of A B are all men over 18 years of age.
Example 2.3. Let S be the space consisting of a real-line segment from 0 to 10
and let A and B be sets of the real-line segments from 1–7 and 3–9 respectively.
Line segments belonging to A [ B, A \ B, A, and B are indicated in Figure 2.4.
Let us note here that, by definition, a set and its complement are always disjoint.
The definitions of union and intersection can be directly generalized to those
involving any arbitrary number (finite or countably infinite) of sets. Thus, the set

n
[
A 1 [ A 2 ... [ A n ˆ A j 2:5†
jˆ1
A
A

0 2 4 6 8 10
B
A ∩ B
A ∪ B

B
Figure 2.4 Sets defined in Example 2.3

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