Using set theory, represent the given voter information. Show this information in a Venn diagram.

Here's the solution to question 4:

Let $U$ be the set of all voters in the voter list.
Let $A$ be the set of people who cast votes for candidate A.
Let $B$ be the set of people who cast votes for candidate B.

Given information:

• Total voters in the voter list, $n(U) = 25,000$.
• People who cast votes for A, $n(A) = 12,000$.
• People who cast votes for B, $n(B) = 10,000$.
• People who cast votes for both A and B, $n(A \cap B) = 1,000$.

a) Write the total number of voters in set notation.
The total number of voters in the voter list is represented by the cardinality of the universal set $U$.

$$\boxed{n(U) = 25,000}$$

b) Show this information in a Venn diagram.
First, calculate the number of people in each region of the Venn diagram:

• Number of people who voted for A only: $n(A \text{ only}) = n(A) - n(A \cap B) = 12,000 - 1,000 = 11,000$
• Number of people who voted for B only: $n(B \text{ only}) = n(B) - n(A \cap B) = 10,000 - 1,000 = 9,000$
• Number of people who voted for at least one candidate (A or B or both): $n(A \cup B) = n(A \text{ only}) + n(B \text{ only}) + n(A \cap B)$ $n(A \cup B) = 11,000 + 9,000 + 1,000 = 21,000$ (Alternatively, $n(A \cup B) = n(A) + n(B) - n(A \cap B) = 12,000 + 10,000 - 1,000 = 21,000$)
• Number of people who did not cast any vote: $n(\text{didn't vote}) = n(U) - n(A \cup B) = 25,000 - 21,000 = 4,000$

Venn Diagram Description:

• Draw a rectangle representing the universal set $U$ with $n(U) = 25,000$.
• Inside the rectangle, draw two overlapping circles. Label one circle $A$ and the other $B$.
• In the overlapping region (intersection $A \cap B$), write 1,000.
• In the part of circle $A$ that does not overlap with $B$ (A only), write 11,000.
• In the part of circle $B$ that does not overlap with $A$ (B only), write 9,000.
• Outside both circles but inside the rectangle, write 4,000.

c) How many people didn't cast vote? Find it.
The number of people who didn't cast a vote is the total number of voters minus those who voted for at least one candidate.

$$n(\text{didn't cast vote}) = n(U) - n(A \cup B)$$ $$n(\text{didn't cast vote}) = 25,000 - 21,000$$ $$n(\text{didn't cast vote}) = 4,000$$

The final answer is $\boxed{4,000}$.

d) Compare the valid vote and invalid vote.