Math Problem Statement

Write the union, intersection, and complement of the given set. Universal Set = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. Set M = {1,3,5,7,9}, Set A = {2,4,6,8,10}, Set T = {11,12,13,14,15}, Set H = {1,2,3,4,5}, Set Z = {6,7,8,9,10}, Set O = {3,5,9,14,15}, Set N = {2,7,8,10,11,13}, Set E = {1,4,5,12,13,14}.

Solution

Let's break down the problem from the image step by step. We have a Universal Set (U) and multiple sets. We are tasked with finding the union, intersection, and complement for the given sets.

Universal Set (U):

U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}

Given Sets:

  • Set M = {1, 3, 5, 7, 9}
  • Set A = {2, 4, 6, 8, 10}
  • Set T = {11, 12, 13, 14, 15}
  • Set H = {1, 2, 3, 4, 5}
  • Set Z = {6, 7, 8, 9, 10}
  • Set O = {3, 5, 9, 14, 15}
  • Set N = {2, 7, 8, 10, 11, 13}
  • Set E = {1, 4, 5, 12, 13, 14}

We need to calculate the following:

  1. Set M∪Set A\text{Set M} \cup \text{Set A}
  2. Set T∪Set H\text{Set T} \cup \text{Set H}
  3. Set Z∪Set E\text{Set Z} \cup \text{Set E}
  4. Set M∩Set A\text{Set M} \cap \text{Set A}
  5. Set M∩Set Z\text{Set M} \cap \text{Set Z}
  6. Set A∩Set O\text{Set A} \cap \text{Set O}
  7. T′T' (Complement of Set T)
  8. H′H' (Complement of Set H)
  9. N′N' (Complement of Set N)
  10. E′E' (Complement of Set E)

Let me calculate these values step by step.Here are the results for the operations involving the given sets:

  1. Set M∪Set A\text{Set M} \cup \text{Set A}: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
  2. Set T∪Set H\text{Set T} \cup \text{Set H}: {1, 2, 3, 4, 5, 11, 12, 13, 14, 15}
  3. Set Z∪Set E\text{Set Z} \cup \text{Set E}: {1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14}
  4. Set M∩Set A\text{Set M} \cap \text{Set A}: ∅\emptyset (no common elements)
  5. Set M∩Set Z\text{Set M} \cap \text{Set Z}: {7, 9}
  6. Set A∩Set O\text{Set A} \cap \text{Set O}: ∅\emptyset (no common elements)
  7. T′T' (Complement of Set T): {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
  8. H′H' (Complement of Set H): {6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
  9. N′N' (Complement of Set N): {1, 3, 4, 5, 6, 9, 12, 14, 15}
  10. E′E' (Complement of Set E): {2, 3, 6, 7, 8, 9, 10, 11, 15}

If you have any questions or would like further details, feel free to ask!

Additional Questions:

  1. What is the difference between union and intersection in set theory?
  2. How do you calculate the complement of a set?
  3. Can two disjoint sets (like M∩AM \cap A and A∩OA \cap O) have any elements in common?
  4. How is the universal set used in calculating complements?
  5. Why is the complement of a set based on the universal set?

Tip:

When calculating the union of two sets, all elements from both sets are included, without repeating any elements.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Union of Sets
Intersection of Sets
Complement of Sets

Formulas

Union of two sets A and B: A ∪ B = {x | x ∈ A or x ∈ B}
Intersection of two sets A and B: A ∩ B = {x | x ∈ A and x ∈ B}
Complement of set A: A' = {x ∈ U | x ∉ A} where U is the universal set

Theorems

De Morgan's Law for Sets: (A ∪ B)' = A' ∩ B', (A ∩ B)' = A' ∪ B'

Suitable Grade Level

Grades 7-9

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