Problem

Quiz 12-1: Basic Set Theory, Counting Outcomes, Probability 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. lements in each: HINT: answers should be like 3,4,5 or what ever numbers would be listed in that section.

Solution

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Text explanation

Given the universal set consists of natural numbers that are at most 16, we first identify all elements of the universal set, then categorize them into Set A (multiples of 3) and Set B (odd numbers), and finally place them accordingly in a Venn diagram.

The elements are $1, 2, 3, \ldots, 16$.

Answer:

Numbers in Each Section:

  • In both A and B (Intersection): $3, 9, 15$
  • Only in A: $6, 12$
  • Only in B: $1, 5, 7, 11, 13$
  • Outside A and B: $2, 4, 8, 10, 14, 16$

Step by Step solution:

Set A (Multiples of 3):

The multiples of 3 up to 16 are $3, 6, 9, 12, 15$.

Set B (Odd Numbers):

The odd numbers up to 16 are $1, 3, 5, 7, 9, 11, 13, 15$.

Placement in the Venn Diagram:

  1. Intersection of A and B (Both multiples of 3 and odd numbers): $3, 9, 15$

  2. Elements only in A (Multiples of 3 but not odd): $6, 12$

  3. Elements only in B (Odd numbers but not multiples of 3): $1, 5, 7, 11, 13$

  4. Elements in neither A nor B (Not multiples of 3 and not odd numbers, so these are even numbers excluding multiples of 3): $2, 4, 8, 10, 14, 16$

Numbers in Each Section:

  • In both A and B (Intersection): $3, 9, 15$
  • Only in A(Multiples of 3 but not odd): $6, 12$
  • Only in B: $1, 5, 7, 11, 13$
  • Outside A and B: $2, 4, 8, 10, 14, 16$
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