Solution Found!
For a finite set A, let N(A) denote the number of elements
Chapter 2, Problem 9STE(choose chapter or problem)
For a finite set A, let N(A) denote the number of elements in A.
(a) Show that
\(N(A \cup B)=N(A)+N(B)-N(AB)\)
(b) More generally, show that
\(\begin{aligned} N\left(\bigcup_{i=1}^n A_i\right)=& \sum_i N\left(A_i\right)-\sum_{i<j} \sum_i N\left(A_i A_j\right)+\cdots+(-1)^{n+1} N\left(A_1 \cdots A_n\right) \end{aligned}\)
Questions & Answers
QUESTION:
For a finite set A, let N(A) denote the number of elements in A.
(a) Show that
\(N(A \cup B)=N(A)+N(B)-N(AB)\)
(b) More generally, show that
\(\begin{aligned} N\left(\bigcup_{i=1}^n A_i\right)=& \sum_i N\left(A_i\right)-\sum_{i<j} \sum_i N\left(A_i A_j\right)+\cdots+(-1)^{n+1} N\left(A_1 \cdots A_n\right) \end{aligned}\)
ANSWER:Step 1 of 2
(a)
For a finite set let
denote the number of elements in
We are asked to prove that
Consider the Venn diagram of events and
Figure 1: Venn diagram in sections
Let us divide into three mutually exclusive sections, as shown in Figure 1. In words, section I represents all the points in
that are not in
(that is,
), section II represents all points both in
and in
(that is,
), and section III represents all points in
that are not in
(that is,
).
From figure 1, we see that
………(1)
As are mutually exclusive, we can write equation (1) as,
Which can be written like this,
Since section II represents all points both in and in
(that is,
)
Hence proved.