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

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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. R1 = {(a, b) : a, b are ages of students and |a – b| 0} R2 = {(a, b) : a, b are weights of students, and |a – b| 0} R3 = {(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

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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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