This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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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 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. 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 only) = n(A) - n(A B) = 12,000 - 1,000 = 11,000 Number of people who voted for B only: n(B only) = n(B) - n(A B) = 10,000 - 1,000 = 9,000 Number of people who voted for at least one candidate (A or B or both): n(A B) = n(A only) + n(B only) + n(A B) n(A B) = 11,000 + 9,000 + 1,000 = 21,000 (Alternatively, n(A B) = n(A) + n(B) - n(A B) = 12,000 + 10,000 - 1,000 = 21,000) Number of people who did not cast any vote: n(didn't vote) = n(U) - n(A 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 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(didn't cast vote) = n(U) - n(A B) n(didn't cast vote) = 25,000 - 21,000 n(didn't cast vote) = 4,000 The final answer is 4,000. d) Compare the valid vote and invalid vote.