Question

Prove the above if provided that n(A) , n ( A U B ) , n(B) are the cardinal number of the following sets .

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Answer 1

$\large\underline{\sf{Solution-}}$

Since,

It is given that,

• A and B are finite sets.

Two cases arises :-

Case 1

• When A and B are not disjoint Sets.

So,

Let assume that,

$\rm :\longmapsto\:n(A - B) = m$

$\rm :\longmapsto\:n(A \: \cap \: B) = n$

$\rm :\longmapsto\:n( B \: - \: A) = p$

Now,

Check the venn diagram, we have,

$\rm :\longmapsto\:n(A \: \cup \:B)$

$\rm \:= \: \:m + n + p$

$\rm \:= \: \:m + n + p + n - n$

$\rm \:= \: \:(m + n) + (n + p) - n$

$\rm \:= \: \:n(A) + n(B) - n(A \: \cap \:B)$

$\bf\implies \:n(A \: \cup \:B)= \: \:n(A) + n(B) - n(A \: \cap \:B)$

Hence, Proved

Again,

$\rm :\longmapsto\:n(A \: \cup \:B)$

$\rm \:= \: \:m + n + p$

$\rm \:= \: \:n(A - B) + n(A \: \cap \:B) + n(B - A)$

Hence, Proved

Case 2

• When A and B are disjoint sets

Since,

It is given that

• A and B are finite sets,

Let assume that,

$\rm :\longmapsto\:n(A) = m$

$\rm :\longmapsto\:n(B) = n$

Now, if we check the venn diagram,

$\rm :\longmapsto\:n(A \: \cup \:B)$

$\rm \:= \: \: m + n$

$\rm \:= \: \: m + n - 0$

$\rm \:= \: \:n(A) + n(B) - n(A \: \cap \:B)$

$\bf\implies \:n(A \: \cup \:B)= \: \:n(A) + n(B) - n(A \: \cap \:B)$

Hence, Proved

Again,

$\rm :\longmapsto\:n(A \: \cup \:B)$

$\rm \:= \: \:m + n$

$\rm \:= \: \:(m - 0) +(n - 0) + 0$

$\rm \:=\bigg(n(A) - n(A\cap B)\bigg) + \bigg(n(B) - n(A\cap B) \bigg) + n(A \cap B)$

$\rm \:= \: \:n(A - B) + n(B - A) + n(A \: \cap \:B)$

Hence, Proved