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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 \( n(U) = n(A \cup B)\)
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.
\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,
\(n(A \cup B \cup C) = n(A) + n(B) + n(C)\)
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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
Mar 28, 2017
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Shajan pandit
In a group of students 20 study account, 21 study maths, 18 study history, 7 study account only, 10 study maths only, 6 study account and maths only, 3 study maths and history only. i) How many students study all subjects?ii) How many students are there altogether?
Dec 31, 2016
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