Question

quiz 12-1: basic set theory, counting outcomes, probability 1-16

1. suppose a universal set consists of natural numbers that are at most 16. two subsets are created from the universal set: set a contains the multiples of 3 and set b contains the odd numbers. correctly place each element of the universal set in the venn diagram to the right.
2. using the sets above, list the elements in each:

a) ( a )
b) ( b )
c) ( a cap b )
d) ( a cup b )
principle to determine the total number of outcomes.

Explanation:

Step1: Define the Universal Set

The universal set \( U \) consists of natural numbers at most 16, so \( U=\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\} \).

Step2: Define Set \( A \) and Set \( B \)

• Set \( A \) (multiples of 3): \( A = \{3, 6, 9, 12, 15\} \)
• Set \( B \) (odd numbers): \( B=\{1,3,5,7,9,11,13,15\} \)

Step3: Find \( A' \) (Complement of \( A \))

\( A' \) includes elements in \( U \) but not in \( A \). So \( A' = \{1,2,4,5,7,8,10,11,13,14,16\} \).

Step4: Find \( B' \) (Complement of \( B \))

\( B' \) includes elements in \( U \) but not in \( B \). So \( B' = \{2,4,6,8,10,12,14,16\} \).

Step5: Find \( A \cap B \) (Intersection of \( A \) and \( B \))

\( A \cap B \) includes elements in both \( A \) and \( B \). So \( A \cap B = \{3,9,15\} \).

Step6: Find \( A \cup B \) (Union of \( A \) and \( B \))

\( A \cup B \) includes elements in \( A \) or \( B \) (or both). So \( A \cup B = \{1,3,5,6,7,9,11,12,13,15\} \).

Answer:

a) \( A' = \{1,2,4,5,7,8,10,11,13,14,16\} \)
b) \( B' = \{2,4,6,8,10,12,14,16\} \)
c) \( A \cap B = \{3,9,15\} \)
d) \( A \cup B = \{1,3,5,6,7,9,11,12,13,15\} \)