Math Problem Statement

Solve the set theory problems involving universal sets, subsets, intersections, unions, complements, and Venn diagrams as presented in the image.

Solution

The image contains mathematical problems related to set theory and Venn diagrams. Let me break down and solve each question from the image.


Question 05

Given:

  • Universal set U={1,2,3,,10}U = \{1, 2, 3, \dots, 10\}
  • Subsets A={2,4,6,9}A = \{2, 4, 6, 9\} and B={1,2,3,4,5}B = \{1, 2, 3, 4, 5\}

We are to find ABA \cap B.

Solution

The intersection of two sets ABA \cap B contains only the elements that are common to both AA and BB: A={2,4,6,9},B={1,2,3,4,5}A = \{2, 4, 6, 9\}, \quad B = \{1, 2, 3, 4, 5\} AB={2,4}A \cap B = \{2, 4\}

Answer: {2,4}\{2, 4\}.


Question 06

Given:

  • Universal set U={a,b,c,d,e}U = \{a, b, c, d, e\}
  • Subsets A={a,e}A = \{a, e\} and B={c,d,e}B = \{c, d, e\}

We are to check which of the given options is incorrect.

Analysis

  1. AB={a,e}{c,d,e}={a,c,d,e}A \cup B = \{a, e\} \cup \{c, d, e\} = \{a, c, d, e\}. This is correct.
  2. AB={a,e}{c,d,e}={e}A \cap B = \{a, e\} \cap \{c, d, e\} = \{e\}. This is correct.
  3. Bc=UB={a,b,c,d,e}{c,d,e}={a,b}B^c = U \setminus B = \{a, b, c, d, e\} \setminus \{c, d, e\} = \{a, b\}. This is correct.
  4. Ac=UA={a,b,c,d,e}{a,e}={b,c,d}A^c = U \setminus A = \{a, b, c, d, e\} \setminus \{a, e\} = \{b, c, d\}. Therefore, n(Ac)=3n(A^c) = 3. This is correct.

Answer: None of the statements are incorrect.


Question 07

Given:

  • Universal set U={1,2,,8}U = \{1, 2, \dots, 8\}
  • Subsets AA and BB such that:
    • AB={4}A \cap B = \{4\}
    • AB={1,3,7}A - B = \{1, 3, 7\}
    • AcBc={2,8}A^c \cap B^c = \{2, 8\}

Find the sum of all elements of BB.

Solution

  1. Using AB={1,3,7}A - B = \{1, 3, 7\}, these elements belong to AA but not BB. Therefore: A={1,3,7,4}A = \{1, 3, 7, 4\}

  2. Using AB={4}A \cap B = \{4\}, BB must include 44.

  3. Using AcBc={2,8}A^c \cap B^c = \{2, 8\}, these elements belong to neither AA nor BB. Thus, the remaining elements of UU are distributed between AA and BB: U={1,2,3,4,5,6,7,8}U = \{1, 2, 3, 4, 5, 6, 7, 8\}

  4. Since A={1,3,4,7}A = \{1, 3, 4, 7\}, Ac={2,5,6,8}A^c = \{2, 5, 6, 8\}. Using AcBc={2,8}A^c \cap B^c = \{2, 8\}: Bc={2,8,1,3,7}B^c = \{2, 8, 1, 3, 7\}

  5. Therefore, B=UBc={4,5,6}B = U \setminus B^c = \{4, 5, 6\}.

Finally, the sum of all elements of BB is: 4+5+6=154 + 5 + 6 = 15

Answer: 1515.


Question 08

Identify the Venn diagram representing A(BCc)A \cap (B \cap C^c).

Solution

  1. BCcB \cap C^c: The region inside BB, but outside CC.
  2. A(BCc)A \cap (B \cap C^c): The overlap of AA with the above region.

The correct diagram is Option ④.


Let me know if you'd like a detailed explanation of any solution or further assistance!


Related Questions

  1. How can we find the complement of a set in set theory problems?
  2. What is the difference between union and intersection of sets?
  3. How do Venn diagrams help in visualizing set operations?
  4. How is the cardinality of a set calculated?
  5. Can you explain how to simplify ABA - B using ABcA \cap B^c?

Tip

To solve set problems efficiently, always write out the elements of each set and carefully use the given operations (union, intersection, complement). Venn diagrams can be a great visual aid!

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Math Problem Analysis

Mathematical Concepts

Set Theory
Universal Set
Subset
Intersection
Union
Complement of a Set
Difference of Sets
Venn Diagram

Formulas

A ∪ B = {x | x ∈ A or x ∈ B}
A ∩ B = {x | x ∈ A and x ∈ B}
A - B = {x | x ∈ A and x ∉ B}
A^c = U \ A (Complement of A with respect to U)

Theorems

De Morgan's Laws
Basic Set Identities
Properties of Intersection and Union

Suitable Grade Level

Grades 9-11