Logics for this lecture:

A relation R on set A is said to be
reflexive if $(a, a) \in R, \forall a \in A {\rm or } aRa, \forall a \in A$
symmetric if $(a, b) \in R \Rightarrow (b, a) \in R, \forall a, b \in A {\rm or } aRb \Rightarrow bRa, \forall a, b \in A$
transitive if $(a, b) \in R, (b, c) \in R \Rightarrow (a, c) \in R, \forall a, b, c \in A {\rm or } aRb {\rm and } bRc, \Rightarrow aRc, \forall a, b, c \in A$
•If a relation is reflexive, symmetric and transitive then the relation is said to be equivalence relation.
• An important property of an equivalence relation is that it divides the set into pairwise disjoint (or mutually disjoint) subsets called equivalence classes whose collection is called a partition of the set.
• The union of all equivalence classes gives the whole set.

## PART - 1 Play Video

Question 1. Determine whether each of the following relations are reflexive, symmetric and transitive:(N)
A. Relation R in the set
A = {1, 2, 3, …, 13, 14} defined as R = {(x, y) : 3x – y = 0}
B. Relation R in the set N of natural numbers defined as R = {(x, y): y = x + 5 and x < 4}
C. Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x}
D. Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer}
E. Relation R in the set A of human beings in a town at a particular time given by
a. R = {(x, y) : x and y work at the same place}
b. R = {(x, y) : x and y live in the same locality}
c. R = {(x, y) : x is exactly 7 cm taller than y}
d. R = {(x, y) : x is wife of y}
e. R = {(x, y) : x is father of y}

## PART - 2 Play Video

Question 2. Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

Question 3. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Question 4. Show that the relation R in the set {1, 2, 3} given by R = { (1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Question 5. Show that each of the relation R in the set A , given by A. R = {(a, b) : |a – b| is a multiple of 4} B. R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Question 6. Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as center.

Question 7. Show that the relation R defined in the set A of all triangles as is equivalence relation. Consider three right angle triangles with sides 3, 4, 5, with sides 5, 12, 13 and with sides 6, 8, 10. Which triangles among are related?

## PART - 3 Play Video

Question 10. Let A = {0, 1, 2, 3, 4} and define a relation R on A as follows: R = {(0, 0), (0, 4), (1, 1), (1, 3), (2, 2), (3, 1), (3, 3), (4, 0), (4, 4)}. As R is an equivalence relation on A. Find the distinct equivalence classes of R.

Question 11. Let R be the relation on set A {0, 1, 2, 3, …, 10} given by R = {(a, b) : (a – b) is divisible by 4}. Show that R is an equivalence relation. Also, write all elements related to 4.

Question 8. Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Question 9. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2 }. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

## PART - 4 Play Video

Question 12. Let A = {1, 2, 3, …. , 12} and R be a relation in A x A defined by (p, q) R (r, s) if ps = qr. Prove that R is an equivalence relation. Also obtain the equivalence class [(3, 4)].

Question 13. Let A = {1, 2, 3, … 9} and R be the relation in A ×A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)].

Question 14. Let N denote the set of all natural numbers and R be the relation on NxN defined by (a, b)R(c, d) if ad(b+c)=bd(a+c) Show that R is an equivalence relation.

ये मंजिलें बड़ी जिद्दी होती हैं, हासिल कहां नसीब से होती हैं।
मगर वहां तूफान भी हार जाते हैं, जहां कश्तियां जिद्द पे होती हैं।।