Math Problem Statement

Question content area top Part 1 Let Uequals​{a, ​b, c,​ d, e,​ f, g,​ h, i,​ j, k},     Aequals​{a​, b​, d​, f​, h​, k​},     Bequals​{b​, d​, f​, h​, j​},     Cequals​{a​, b​, g​, j​, k​}. Determine Upper A union left parenthesis Upper C intersect Upper B right parenthesis prime.

Solution

To find A(CB)A \cup (C \cap B)', let's follow the steps:

  1. Find CBC \cap B:

    • C={a,b,g,j,k}C = \{ a, b, g, j, k \}
    • B={b,d,f,h,j}B = \{ b, d, f, h, j \}
    • CB={b,j}C \cap B = \{ b, j \}
  2. Find (CB)(C \cap B)':

    • The universal set U={a,b,c,d,e,f,g,h,i,j,k}U = \{ a, b, c, d, e, f, g, h, i, j, k \}
    • (CB)(C \cap B)' includes all elements in UU that are not in CBC \cap B:
    • (CB)={a,c,d,e,f,g,h,i,k}(C \cap B)' = \{ a, c, d, e, f, g, h, i, k \}
  3. Find A(CB)A \cup (C \cap B)':

    • A={a,b,d,f,h,k}A = \{ a, b, d, f, h, k \}
    • (CB)={a,c,d,e,f,g,h,i,k}(C \cap B)' = \{ a, c, d, e, f, g, h, i, k \}
    • A(CB)={a,b,c,d,e,f,g,h,i,k}A \cup (C \cap B)' = \{ a, b, c, d, e, f, g, h, i, k \}

Thus, A(CB)={a,b,c,d,e,f,g,h,i,k}A \cup (C \cap B)' = \{ a, b, c, d, e, f, g, h, i, k \}.

Would you like more details or have any questions about this solution? Here are some related questions:

  1. What are the elements of ABA \cap B?
  2. Can you explain how to find the complement of a set?
  3. What is the significance of unions and intersections in set theory?
  4. How would you represent these sets using Venn diagrams?
  5. What would change if the universal set were different?

Tip: Always double-check the elements when finding intersections or complements!

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Math Problem Analysis

Mathematical Concepts

Set Theory
Set Operations
Complement of a Set

Formulas

A ∪ B
A ∩ B
A'

Theorems

De Morgan's Laws

Suitable Grade Level

Grades 9-11