Math Problem Statement

Let A = {1, 2, 4, 7, 8}, B = {1, 4, 5, 7, 9}, and C = {3, 7, 8, 9}. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find set expressions using these sets and the operations of union, intersection, absolute difference, and relative difference to represent the sets: (a) {2, 7, 9} (b) {3, 5, 6, 7, 9, 10}.

Solution

To solve this problem, we need to express each target set using the provided sets AA, BB, CC, and the universal set UU, with operations including union, intersection, absolute difference, and relative difference.

Given sets:

  • A={1,2,4,7,8}A = \{1, 2, 4, 7, 8\}
  • B={1,4,5,7,9}B = \{1, 4, 5, 7, 9\}
  • C={3,7,8,9}C = \{3, 7, 8, 9\}
  • U={1,2,3,4,5,6,7,8,9,10}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

We need to find expressions for the sets:

  1. (a) {2,7,9}\{2, 7, 9\}
  2. (b) {3,5,6,7,9,10}\{3, 5, 6, 7, 9, 10\}

Solution

(a) Set {2,7,9}\{2, 7, 9\}

  1. Identify common elements in sets:

    • 22 is in AA.
    • 77 is in AA, BB, and CC.
    • 99 is in BB and CC.
  2. Construct an expression:

    • Since 77 is common to all three sets, we need to isolate 22 and 99 appropriately. We can start by considering intersections and relative differences.
    • A possible approach: (AB)C(A \cup B) \cap C

This will yield the set {2,7,9}\{2, 7, 9\}.

(b) Set {3,5,6,7,9,10}\{3, 5, 6, 7, 9, 10\}

  1. Identify common elements:

    • 33 and 99 are in CC.
    • 55 and 77 are in BB.
    • 66 and 1010 are only in the universal set UU but not in AA, BB, or CC.
  2. Construct an expression:

    • Start by considering the elements that are not in AA but are in UU, which can be represented as UAU - A.

    Thus, we can write the expression as: UA={3,5,6,7,9,10}U - A = \{3, 5, 6, 7, 9, 10\}

Final Answers

  • (a) {2,7,9}=(AB)C\{2, 7, 9\} = (A \cup B) \cap C
  • (b) {3,5,6,7,9,10}=UA\{3, 5, 6, 7, 9, 10\} = U - A

Would you like further explanation on these operations or have additional questions?


Here are five related questions to explore further:

  1. How would you find the union of all three sets AA, BB, and CC?
  2. What is the intersection of sets AA and BB?
  3. How would you represent ABA - B using set operations?
  4. How could we determine the symmetric difference between sets BB and CC?
  5. What is the complement of CC with respect to the universal set UU?

Tip: When working with multiple sets and set operations, it can be helpful to visualize them using Venn diagrams for a clearer understanding of relationships.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Set Theory
Union and Intersection of Sets
Relative Difference
Absolute Difference

Formulas

Union of sets: A ∪ B
Intersection of sets: A ∩ B
Relative difference of sets: A - B
Absolute difference between sets

Theorems

-

Suitable Grade Level

Grades 9-12