Chapter 1 - Types of Relations (PYQs)

Write the smallest reflexive relation on set $A = \{a, b, c\}$.

A relation R in a set A is called ______ if (a₁, a₂) ∈ R implies (a₂, a₁) ∈ R, for all a₁, a₂ ∈ A.

A relation R in a set A is called ______ relation, if each element of A is related to itself.

Let A = {3, 5}. Then number of reflexive relations on A is (a) 2 (b) 4 (c) 0 (d) 8

The number of equivalence relations in the set {1, 2, 3} containing the elements (1, 2) and (2, 1) is (a) 0 (b) 1 (c) 2 (d) 3

The relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1), (1, 1)} is (a) symmetric and transitive, but not reflexive (b) reflexive and symmetric, but not transitive (c) symmetric, but neither reflexive nor transitive (d) an equivalence relation

Let R be a relation in the set N given by R = {(a, b) : a = b − 2, b > 6}. Then (a) (8, 7) ∈ R (b) (6, 8) ∈ R (c) (3, 8) ∈ R (d) (2, 4) ∈ R

A relation R is defined on N. Which of the following is reflexive relation? (a) R = {(x, y) : x > y, x, y ∈ N} (b) R = {(x, y) : x + y = 10, x, y ∈ N} (c) R = {(x, y) : x is the square number, x, y ∈ N} (d) R = {(x, y) : x + 4y = 10, x, y ∈ N}

Check if the relation R in the set R of real numbers defined as R = {(a, b) : a < b} is (i) symmetric (ii) transitive.

Let W denote the set of words in the English dictionary. Define the relation R by R = {(x, y) ∈ W × W such that x and y have at least one letter in common}. Show that this relation R is reflexive and symmetric, but not transitive.

Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R on the set Z of all integers, given by R = {(a, b) : 2 divides (a − b)} is an equivalence relation.

Show that the relation R on R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation S in the set of real numbers R defined by S = {(a, b) : where a − b + √2 is an irrational number} is reflexive, symmetric or transitive.

Show that the relation S in set R of real numbers defined by S = {(a, b) : a ≤ b³, a ∈ R, b ∈ R} is neither reflexive, nor symmetric, nor transitive.

Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Hence, find the elements of equivalence class [1].

A relation R is defined on a set of real numbers R as R = {(x, y) : x − y is an irrational number}. Check whether R is reflexive, symmetric and transitive or not.

If N denotes the set of all natural numbers and R is the relation on N × N defined by (a, b) R (c, d), if ad(b + c) = bc(a + d). Show that R is an equivalence relation.

How many relations are possible from B to G?

Among all the possible relations from B to G, how many functions can be formed from B to G?

Case Study: A class-room teacher is keen to assess the learning of her study concept of “relations” taught to them. She writes the following five relations each defined on the set A = {1, 2, 3}: R₁ = {(2, 3), (3, 2)}, R₂ = {(1, 2), (1, 3), (3, 2)}, R₃ = {(1, 2), (2, 1), (1, 1)}, R₄ = {(1, 1), (1, 2), (3, 3), (2, 2)}, R₅ = {(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)}. (i) Identify the relation which is reflexive, transitive but not symmetric. (ii) Identify the relation which is reflexive and symmetric but not transitive. (iii) (a) Identify the relations which are symmetric but neither reflexive nor transitive. OR (iii) (b) What pairs should be added to the relation R₂ to make it an equivalence relation?

Case Study: Students of a school are taken to a railway museum. Let L be the set of all rail lines and R = {(l₁, l₂) : l₁ is parallel to l₂}. (i) Find whether R is symmetric. (ii) Find whether R is transitive. (iii) If one line is y = 3x + 2, find all lines in R related to it. OR Let S = {(l₁, l₂) : l₁ is perpendicular to l₂}. Check whether S is symmetric and transitive.

Show that the relation R in the set A = {1, 2, 3, 4, 5, 6} given by R = {(a, b) : |a − b| is divisible by 2} is an equivalence relation.

Show that the relation S in the set A = {x ∈ Z : 0 ≤ x ≤ 12} given by S = {(a, b) : |a − b| is divisible by 3} is an equivalence relation.

A relation R is defined on N × N (where N is the set of natural numbers) as (a, b) R (c, d) ⇔ a − c = b − d. Show that R is an equivalence relation.

A relation R is defined on N × N (where N is the set of natural numbers) as (a, b) R (c, d) ⇔ a/b = c/d. Show that R is an equivalence relation.

Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b) : a, b ∈ A, |a − b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].

Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Hence, find the elements of equivalence class [1].