Chapter 1: Relations

Relations

2. Relations

Ordered Pairs

Definition: An ordered pair is a pair of elements written in a specific order, denoted by (a, b).

Equality of Ordered Pairs: Two ordered pairs (a, b) and (c, d) are equal if and only if a = c and b = d.

Example:

(2, 5) = (2, 5) is true because both the first and second elements match.

(2, 5) ≠ (5, 2) because the order of elements matters.

Exercise:

Determine whether (3, 7) = (7, 3).

Cartesian Product of Relations

Definition: Given two relations (sets) A and B, their Cartesian product is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Notation: A × B = { (a,b) | a ∈ A, b ∈ B }

Example:

Let A = {1, 2} and B = {x, y}

Then A × B = { (1,x), (1,y), (2,x), (2,y) }

Set A	Set B	Cartesian Product A × B
{1, 2}	{x, y}	{(1,x), (1,y), (2,x), (2,y)}

Exercise:

Find A × B if A = {3, 4} and B = {a, b}.

Number of Elements in the Cartesian Product

If n(A) is the number of elements in set A and n(B) in set B, then:

Formula: n(A × B) = n(A) × n(B)

Example:

If n(A) = 3 and n(B) = 4, then:

n(A × B) = 3 × 4 = 12

Exercise:

Find the number of elements in A × B if A has 5 elements and B has 2 elements.

Cartesian Product of the Set of Reals with Itself

Definition: The Cartesian product ℝ × ℝ is the set of all ordered pairs of real numbers.

Each point (x, y) represents a unique location on a two-dimensional plane.

Diagram: Cartesian Plane showing (x,y) points.

Exercise:

Give two examples of points in ℝ × ℝ.

Definition of Relation

A relation R from a set A to a set B is a subset of A × B.

If (a, b) ∈ R, we say that “a is related to b” under relation R.

Example:

If A = {1,2} and B = {3,4}, a possible relation R could be {(1,3), (2,4)}

Exercise:

Write any two relations from A = {1,2} to B = {a,b,c}.

Pictorial Diagrams of Relations

Relations can be shown using arrow diagrams where elements of set A are linked to elements of set B.

Diagram: Visual representation of a Relation.

Exercise:

Draw an arrow diagram for A = {1,2,3} and B = {x,y} where 1 → x, 2 → y, 3 → x.

Domain, Co-domain, and Range of a Relation

Domain: Set of all first elements in the ordered pairs.

Co-domain: Set B (second set).

Range: Actual second elements related to elements of A.

Term	Meaning	Example
Domain	First elements	{1, 2}
Co-domain	Second set (target)	{x, y, z}
Range	Actually related elements	{x, y}

Exercise:

Identify the domain, co-domain, and range for the relation R = {(1, a), (2, b)} where A = {1,2,3} and B = {a,b,c}.

Practice Test: Relations

Define an ordered pair and give an example.
If A = {1,2} and B = {3,4}, list A × B.
Find the number of elements in A × B if A has 2 elements and B has 5 elements.
Give an example of a relation from A = {a,b} to B = {1,2}.
Draw an arrow diagram for the relation R = {(a,1), (b,2)}.
Identify the domain, co-domain, and range for the relation {(2,5), (3,7)}.

Relations – MCQs Practice

Practice MCQs: Relations

1. If A = {1,2,3} and B = {x,y}, how many relations are possible from A to B?

a) 64 b) 128 c) 36 d) 512

2. How many reflexive relations are possible on a set of 3 elements?

a) 8 b) 64 c) 512 d) 2⁶

3. The relation R = {(a,a) | a ∈ A} is always:

a) Symmetric only b) Reflexive only c) Reflexive and Symmetric d) None

4. The number of symmetric relations on a set with 4 elements is:

a) 2¹⁰ b) 2⁶ c) 2⁸ d) 2¹⁶

5. The relation R = {(x,y) | x,y ∈ ℝ, x² + y² = 1} is:

a) Reflexive b) Symmetric c) Transitive d) None

6. In how many ways can you form a relation on a set of 2 elements?

a) 4 b) 8 c) 16 d) 32

7. If n(A) = 5 and n(B) = 3, then the number of functions from A to B is:

a) 5³ b) 3⁵ c) 8 d) 15

8. A relation R on a set A is said to be transitive if:

a) (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R b) (a,b) ∈ R ⇒ (b,a) ∈ R c) (a,a) ∈ R d) None of these

9. A relation R = {(a,b) | a divides b} on ℕ is:

a) Reflexive b) Symmetric c) Anti-symmetric d) Both (a) and (c)

10. Let A = {1,2}, the number of equivalence relations on A is:

a) 1 b) 2 c) 3 d) 4

11. If a relation is reflexive and symmetric but not transitive, it is called:

a) An equivalence relation b) A partial equivalence c) A tolerance relation d) None of these

12. The relation “less than” (<) on ℝ is:

a) Reflexive b) Symmetric c) Anti-symmetric d) None

13. Which of the following relations are always symmetric?

a) Identity relation b) Universal relation c) Empty relation d) All of these

14. If R = {(1,2), (2,3), (3,1)}, is R transitive?

a) Yes b) No c) Can’t say d) None

15. Which relation is both symmetric and transitive?

a) {(a,a), (b,b)} b) {(a,b), (b,a)} c) {(a,b), (b,a), (b,b)} d) {(a,a)}

Answers

Q1	Q2	Q3	Q4	Q5
a	d	c	b	b
Q6	Q7	Q8	Q9	Q10
c	b	a	d	3
Q11	Q12	Q13	Q14	Q15
c	d	d	b	d

Self Assessment Test – Relations

Self Assessment Test: Relations

Section A: Short Answer Questions (2-3 lines each)

1. Define an ordered pair with an example.

2. Find the number of elements in the Cartesian product A × B if n(A) = 3 and n(B) = 4.

3. What is the domain and range of the relation R = {(2,3), (3,5), (4,7)}?

4. Define reflexive relation with one example.

5. Give an example of a relation which is symmetric but not transitive.

Section B: Long Answer Questions (Detailed)

1. Define Cartesian product of two sets. Explain its properties with a proper diagram.

2. Explain domain, co-domain, and range with examples. Differentiate between co-domain and range.

3. If A = {1, 2, 3} and B = {a, b}, list all elements of A × B. Also, define a relation R from A to B such that the first element is an odd number.

4. Show that the relation R on the set of real numbers defined by (a, b) ∈ R if and only if a – b is an integer, is an equivalence relation.

Answers

Section A Q1	Section A Q2	Section A Q3	Section A Q4	Section A Q5
Defined + Example	12	Domain = {2,3,4}, Range = {3,5,7}	Definition + Example	Example based
Section B Q1	Section B Q2	Section B Q3	Section B Q4
Definition + Properties	Explanation + Difference	List all + Define R	Proved as Equivalence
Section C Q1	Section C Q2	Section C Q3	Section C Q4	Section C Q5
b	b	b	c	b

Relations

2. Relations

Ordered Pairs

Example:

Exercise:

Cartesian Product of Relations

Example:

Exercise:

Number of Elements in the Cartesian Product

Example:

Exercise:

Cartesian Product of the Set of Reals with Itself

Exercise:

Definition of Relation

Example:

Exercise:

Pictorial Diagrams of Relations

Exercise:

Domain, Co-domain, and Range of a Relation

Exercise:

Practice Test: Relations

Practice MCQs: Relations

1. If A = {1,2,3} and B = {x,y}, how many relations are possible from A to B?

2. How many reflexive relations are possible on a set of 3 elements?

3. The relation R = {(a,a) | a ∈ A} is always:

4. The number of symmetric relations on a set with 4 elements is:

5. The relation R = {(x,y) | x,y ∈ ℝ, x² + y² = 1} is:

6. In how many ways can you form a relation on a set of 2 elements?

7. If n(A) = 5 and n(B) = 3, then the number of functions from A to B is:

8. A relation R on a set A is said to be transitive if:

9. A relation R = {(a,b) | a divides b} on ℕ is:

10. Let A = {1,2}, the number of equivalence relations on A is:

11. If a relation is reflexive and symmetric but not transitive, it is called:

12. The relation “less than” (<) on ℝ is:

13. Which of the following relations are always symmetric?

14. If R = {(1,2), (2,3), (3,1)}, is R transitive?

15. Which relation is both symmetric and transitive?

Answers

Self Assessment Test: Relations

Section A: Short Answer Questions (2-3 lines each)

Section B: Long Answer Questions (Detailed)

Section C: Case Study Based MCQs

Answers