**Logics for this lecture: **

*Relation \subseteq Cartesian Product**Relation on set A is defined as R \subseteq A \times A ** \phi and A \times A are two extreme subsets of A \times A **If R = \phi , then relation is called empty relation.**If R = A \times A , then **relation is called universal relation.**Both empty and universal relations are also known as trivial relations.*

## PART - 1

**Question 1.** Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b 6}. Choose the correct answer. A. (2, 4) ∈ R B. (3, 8) ∈ R C. (6, 8) ∈ R D. (8, 7) ∈ R **(N)**

**Question 2.** Let A= {1, 2, 3,} and define R = {(a, b): a – b = 12}. Show that R is empty relation on Set A. **(B)**

**Question 3.** Let A be the set of all students of a boy’s school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R′ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. **(N)**

**Question 4.** If A is the set of students of a school then write, which of following relations are Universal, Empty or neither of the two. R_{1} = {(a, b) : a, b are ages of students and |a – b| 0} R_{2} = {(a, b) : a, b are weights of students, and |a – b| 0} R_{3} = {(a, b) : a, b are students studying in same class} **(B)**

**Question 5.** Let A = {1, 2, 3,} and define R = {(a, b): a + b 0}. Show that R is a universal relation on set A. **(B)**

## PART - 2

**Question 6.** Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive? **(E)**

**Question 7.** Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer. **(N)** **A.** R is reflexive and symmetric but not transitive. **B.** R is reflexive and transitive but not symmetric. **C.** R is symmetric and transitive but not reflexive. **D.** R is an equivalence relation.

**Question 8.** Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive. **(N)**

**Question 9.** Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive. **(N)**

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