In an election of municipality, A and B were two candidates for the post of the mayor and 25,000 voters were in the voter list. Voters were supposed to cast the vote for a single candidate. 12,000 people cast cote for A, 10,000 people cast for B and 1,000 people cast even for both the candidates.

1. Show these information in a Venn-diagram.
2. How many people didn't cast the vote?
3. How many votes were valid?

Solution:

Let U be the set of total votes casted in the election.

Let A and B be the sets of people who casted vote for candidates A and B, respectively.

According to the question,

$n(U) = 25000$

$n_o(A) = 12000$

$n_o(B) = 10000$

$n(A \cap B) = 1000$

Using formula,

$n(U) = n_o(A) + n_o(B) + n(A \cap B) + n(A \cup B)^c$

$or, 25000 = 12000 + 10000 + 1000 + n(A \cup B)^c$

$or, 25000 = 23000 + n(A \cup B)^c$

$or, n(A \cup B)^c = 25000 - 23000$

$\therefore n(A \cup B)^c = 2000$

So, 2,000 people did not cast votes in the election.

And,

$n_o(A) + n_o(B) = 12000 + 10000$

$= 22000$

Hence, 22,000 total valid votes were cast in the election.

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