## Note on Cardinality of a set

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The cardinality of set A is defined as the number of elements in the set A and is denoted by n(A).
For example, if A = {a,b,c,d,e} then cardinality of set A i.e.n(A) = 5

Let A and B are two subsets of a universal set U. Their relation can be shown in Venn-diagram as:

$$n(A) = n_o( A) + n(A \cap B)$$

$$\text{or,}\: n(A) - n (A \cap B)= n_o(A)$$

$$n(B) = n_o(B) + n(A \cap B)$$

$$\text {or,}\: n(B) - n(A \cap B) = n_o(B)$$

Also,

\begin{align*} n(A∪B) &= n_o(A) + n(A∩B) + n_o(B)\\ n(A∪B) &= n(A) - n(A∩B) + n(A∩B) + n(B) - n(A∩B)\\ n(A∪B) &= n(A) + n(B)- n(A∩B)\\ \therefore n(A∪B) &= n(A) + n(B) - n(A∩B)\\ \end{align*}

If A and B are disjoint sets then:

$$n(A \cap B) = 0, n(A \cup B) =n(A) + n(B)$$

Again,

$$n(U) = n(A \cup B) + n(\overline {A\cup B)}$$

If $$n(\overline {A \cup B)}$$=0, then $$n(U) = n(A \cup B)$$

#### Problems involving three sets

Let A, B and C are three non-empty and intersecting sets, then:
$$n(A \cup B \cup C) = n(A) + n(B) +n(C) - n(A \cap B) -n(B \cap C) -n(C \cap A) +n(A \cap B \cap C).$$

In Venn-diagram

$$n(A)$$ = Number of elements in set A.

$$n(B)$$ = Number of elements in set B.

$$n(C)$$=Number of element in set C.

$$n_o(A)$$ = Number of elements in set A only.

$$n_o(B)$$ = Number of elements in set B only.

$$n_o(C)$$ = Number of elements in set C only.

$$n_o(A \cap B)$$ = Number of elements in set A and B only.

$$n_o(B \cap C)$$ = Number of elements in set B and C only.

$$n_o(C \cap A)$$ = Number of elements in set A and C only.

$$n(A \cap B \cap C)$$ = Number of elements in set A, B and C.

#### From the Venn-diagram

\begin{align*} n(A \cup B \cup C) &= n_o(A) +n_o(B) +n_o(C) +n_o(A \cap B) +n_o(B \cap C) +n_o(C \cap A) + n(A \cap B \cap C)\\ &= n(A) - n_o(A \cap B) - n_o(C \cap A) - n(A \cap B \cap C) + n(B) - n_o(B \cap C) - n_o(C\cap B) - n(A \cap B \cap C)
+ n(C) - n_o(A \cap C) - n_o(B \cap C) - n(A \cap B \cap C) + n_o(A \cap B) +n_o(B \cap C) +n_o(C \cap A) + n(A \cap B \cap C)\\ &= n(A) + n(B) + n(C) - [n_o(A \cap B) +n(A \cap B \cap C)] - [n_o(A \cap B) +n(A \cap B \cap C)] - [n_o(B \cap C) +n(A \cap B \cap C)] - [n_o(C \cap A) +n(A \cap B \cap C)]+n(A \cap B \cap C)\\ &= n(A) + n(B) + n(C) - n(A \cap B) -n(B \cap C) -n(A \cap C) +n(A \cap B \cap C)\\ \end{align*}

$$\boxed{\therefore (A \cup B \cup C)= n(A) + n(B) + n(C) - n(A \cap B) -n(B \cap C) -n(C \cap A) +n(A \cap B \cap C)}$$

If A, B and C are disjoint sets,

$$n(A \cup B \cup C) = n(A) + n(B) + n(C)$$

• The cardinality of a set is a positive integer but it is not decimal. So, n(A)  is not equal to 50% because  50% = 0.5.
• If A, B and C are disjoint sets, $$n(A \cup B \cup C) = n(A) + n(B) + n(C).$$
• $$(A \cup B \cup C)= n(A) + n(B) + n(C) - n(A \cap B) -n(B \cap C) -n(C \cap A) +n(A \cap B \cap C)$$
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##### Shrestha khatri

A, B and C are the subset of U. If n(U) =130,n(AnB)=150 and no.( C) = 15 find n( AuBuC) and (AuBuC)completment