Set Operations Quiz: Master Your Skills

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Questions and Answers

Write the set {−1, 1} in set builder form.

{x ∈ Z : x = -1 or x = 1}

State whether the given set is finite or infinite: {x ∈ N : x is an even prime number}.

Infinite

Prove that ((AÈB’ÈC)∩(A∩B’∩C’))È((AÈBÈC’)∩(B’∩C’)) = B’∩C

This is a proof-based question and requires a detailed explanation.

Find the number of sets B ⊆ X such that A − B = {4}.

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If n(P(A)) = 1024, n(A ∪ B) = 15 and n(P(B)) = 32, then find n(A ∩ B).

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Find the number of subsets of A if A = {x : x = 4n + 1, 2 ≤ n ≤ 5, n ∈ N}.

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If n(A ∩ B) = 3 and n(A È B) = 10, find n(P(AΔB)).

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If A = {1, 2, 3} and B = {2, 4, 6}, find A ´ B.

{(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6)} Signup and view all the answers

For a set A, A ´ A contains 16 elements, two of its elements are (1, 3) and (0, 2). Find elements of A.

{0, 1, 2, 3} Signup and view all the answers

Let A = {1,2,3}, B = {3,4} and C = {4,5,6}. Find i) A × (B Ç C) ii) (A × B) Ç (A × C) iii) A × (B È C) iv) (A × B) È (A × C).

i) {(1, 4)} ii) {} iii) {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)} iv) {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)} Signup and view all the answers

If A and B are two sets so that n(B − A) = 2n(A − B) = 4n(A ∩ B) and if n(A È B) = 14, then find n(P(A)).

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Prove that relation friendship is not an equivalence relation on the set of people in Chennai.

The relation of friendship is not reflexive, symmetric, and transitive on the set of people in Chennai. Hence, it is not an equivalence relation. Signup and view all the answers

Let X = {1, 2, 3, 4} R = {(1, 1), (2, 1), (2, 2), (3, 3),(1, 3),(4,4),(1,2),(3, 1)}. Is the relation R reflexive, symmetric, and transitive?

Reflexive: No, Symmetric: No, Transitive: No Signup and view all the answers

On the set of natural numbers R be the relation defined by xRy if x + 2y = 21. Is the relation R reflexive, symmetric, and transitive?

Reflexive: No, Symmetric: No, Transitive: Yes Signup and view all the answers

On the set of natural numbers, the relation R is defined by “mRn if m divides n”. Is the relation R reflexive, symmetric, and transitive?

Reflexive: Yes, Symmetric: No, Transitive: Yes Signup and view all the answers

Let X = {1, 2, 3, 4} and R = f, empty set. Is the relation R = {(1,1),(2,2),(3,3),...,(n, n)} reflexive, symmetric, and transitive?

Reflexive: Yes, Symmetric: No, Transitive: Yes Signup and view all the answers

Study Notes

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).

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Test your knowledge of sets and set operations with this quiz featuring questions about writing sets in roster and set builder form, determining the finiteness or infiniteness of given sets, and more.

Sets and Set Operations Quiz

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