Maths Part 1 Revision

Detailed Concepts with Examples

1. Union and Intersection of Sets

Union ( ∪ ): The union of two sets A and B is a set containing all elements that are in A, in B, or in both.

Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.

Intersection ( ∩ ): The intersection of two sets A and B is a set containing all elements that are in both A and B.

Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}.

2. Symmetric and Anti-Symmetric Relations

Symmetric Relation: A relation R on a set A is symmetric if (a, b) ∈ R implies (b, a) ∈ R.

Example: If A = {1, 2} and R = {(1, 2), (2, 1)}, then R is symmetric.

Anti-Symmetric Relation: A relation R on a set A is anti-symmetric if (a, b) ∈ R and (b, a) ∈ R implies a = b.

Example: If A = {1, 2, 3} and R = {(1, 1), (2, 2), (1, 2)}, then R is anti-symmetric.

3. Equivalence Relation

An equivalence relation is a relation that is reflexive, symmetric, and transitive.

Example: "is equal to" is an equivalence relation because it is reflexive (a = a), symmetric (if a = b, then b = a), and transitive (if a = b and b = c, then a = c).

Key Concepts and Formulas

Union: \( A ∪ B = \{ x | x \in A \text{ or } x \in B \} \)
Intersection: \( A ∩ B = \{ x | x \in A \text{ and } x \in B \} \)
Symmetric Relation Condition: \( \forall a, b \in A, (a, b) \in R \implies (b, a) \in R \)
Anti-Symmetric Relation Condition: \( \forall a, b \in A, (a, b) \in R \text{ and } (b, a) \in R \implies a = b \)
Equivalence Relation Conditions:

10 Multiple-Choice Questions (MCQs)

1. What is the union of sets A = {1, 2} and B = {2, 3}?

a) {1, 2}

b) {2, 3}

c) {1, 2, 3}

d) {1, 3}

Answer: c) {1, 2, 3}

2. What is the intersection of sets A = {1, 2} and B = {2, 3}?

a) {1}

b) {2}

c) {3}

d) {1, 3}

Answer: b) {2}

3. Which of the following is a symmetric relation?

a) {(1, 2), (2, 1)}

b) {(1, 2), (2, 3)}

c) {(1, 1), (2, 2)}

d) {(1, 2), (3, 4)}

Answer: a) {(1, 2), (2, 1)}

4. Which of the following is an anti-symmetric relation?

a) {(1, 2), (2, 1)}

b) {(1, 1), (2, 2)}

c) {(1, 2), (2, 2)}

d) {(1, 2), (2, 3)}

Answer: b) {(1, 1), (2, 2)}

5. An equivalence relation must be:

a) Reflexive and Symmetric

b) Symmetric and Transitive

c) Reflexive, Symmetric, and Transitive

d) Reflexive and Transitive

Answer: c) Reflexive, Symmetric, and Transitive

6. If a relation R on set A is reflexive, which of the following is true?

a) (a, b) ∈ R

b) (b, a) ∈ R

c) (a, a) ∈ R

d) (a, c) ∈ R

Answer: c) (a, a) ∈ R

7. Which of the following is an example of a relation that is both symmetric and anti-symmetric?

a) {(1, 1), (2, 2)}

b) {(1, 2), (2, 1)}

c) {(1, 2), (2, 3)}

d) {(1, 2), (3, 1)}

Answer: a) {(1, 1), (2, 2)}

8. The intersection of sets A and B is empty. Which of the following statements is true?

a) A ∩ B = A

b) A ∩ B = B

c) A ∩ B = ∅

d) A ∩ B = A ∪ B

Answer: c) A ∩ B = ∅

9. Which of the following statements is true about the union of sets A and B?

a) A ∪ B contains only elements that are in both A and B.

b) A ∪ B contains all elements that are in either A or B or both.

c) A ∪ B is always equal to A.

d) A ∪ B is always equal to B.

Answer: b) A ∪ B contains all elements that are in either A or B or both.

10. If R is an equivalence relation on set A, then:

a) R is only reflexive.

b) R is only symmetric.

c) R is only transitive.

d) R is reflexive, symmetric, and transitive.

Answer: d) R is reflexive, symmetric, and transitive.