Sets and Set Operations Quiz

Set Theory

• The set {−1, 1} can be written in set builder form as {x : x = -1 or x = 1}.
• The set {x ∈ N : x is an even prime number} is finite, containing only the element 2.

Set Operations and Identities

• ((AÈB’ÈC)∩(A∩B’∩C’))È((AÈBÈC’)∩(B’∩C’)) = B’∩C (proven identity)
• If A - B = {4}, then the number of sets B ⊆ X is dependent on the elements of X.

Power Sets and Cardinality

• If n(P(A)) = 1024, n(A ∪ B) = 15, and n(P(B)) = 32, then n(A ∩ B) = 2 (solved example)
• If A = {x : x = 4n + 1, 2 ≤ n ≤ 5, n ∈ N}, then the number of subsets of A is 2^4 = 16.
• If n(A ∩ B) = 3 and n(A È B) = 10, then n(P(AΔB)) = 2^10 = 1024.

Cartesian Products

• If A = {1, 2, 3} and B = {2, 4, 6}, then A ´ B = {(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6)}.
• If A ´ A contains 16 elements, including (1, 3) and (0, 2), then A = {0, 1, 2, 3}.
• If A = {1,2,3}, B = {3,4}, and C = {4,5,6}, then:
  • A × (B Ç C) = {(1, 4), (2, 4), (3, 4)}
  • (A × B) Ç (A × C) = {(3, 4), (3, 4)}
  • A × (B È C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}
  • (A × B) È (A × C) = {(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)}

Relations and Functions

• The relation "friendship" is not an equivalence relation on the set of people in Chennai (proof required).
• The relation R = {(1, 1), (2, 1), (2, 2), (3, 3), (1, 3), (4, 4), (1, 2), (3, 1)} on X = {1, 2, 3, 4} is:
  • Reflexive: yes, since (1, 1), (2, 2), (3, 3), and (4, 4) are in R.
  • Symmetric: no, since (1, 2) is in R but (2, 1) is not.
  • Transitive: no, since (1, 2) and (2, 3) are in R, but (1, 3) is not.
• The relation R defined by xRy if x + 2y = 21 on the set of natural numbers is not reflexive, symmetric, or transitive (proof required).
• The relation R defined by "mRn if m divides n" on the set of natural numbers is:
  • Reflexive: yes, since every number divides itself.
  • Symmetric: no, since if m divides n, it does not imply that n divides m.
  • Transitive: yes, since if m divides n and n divides p, then m divides p.
• The relation R = {(1,1),(2,2),(3,3),...,(n, n)} on any set is:
  • Reflexive: yes, since every element is related to itself.
  • Symmetric: yes, since (a, a) is in R implies (a, a) is in R.
  • Transitive: yes, since (a, a) and (a, a) implies (a, a).