1 You Never Escape Your… Relations

EQUIVALENCE RELATION Presented by K.Senguttuvan, PGT Kendriya Vidyalaya, Gachibowli, Hyderabad. 2

3 RELATIONS A relation R from a non-empty set A to a non-empty B is a subset of the Cartesian product A X B. A relation R from a non-empty set A to a non-empty B is a subset of the Cartesian product A X B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A X B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A X B.

TYPES OF RELATIONS Empty Relation Universal Relation Reflexive Relation Symmetric Relation Transitive Relation 4

5 EMPTY RELATION A Relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = A X A. Example 1 : consider the relation R on the set A ={1, 2, 3} given by R={(a, b):a, b Є A, a + b=10} R={(a, b):a, b Є A, a + b=10} so, A XA = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.

6 UNIVERSAL RELATION A Relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A X A. i.e., R = A X A. Example 2 : consider the relation R on the set A = {1, 2, 3} given by Example 2 : consider the relation R on the set A = {1, 2, 3} given by R = {(a, b): a, b Є A, |a-b| ≥ 0}. R = {(a, b): a, b Є A, |a-b| ≥ 0}. we note that all pairs (a, b) in A X A satisfy |a-b| ≥ 0 we note that all pairs (a, b) in A X A satisfy |a-b| ≥ 0 so, R is the whole set A X A. so, R is the whole set A X A. NOTE : Both empty relation and universal relation are sometimes called trivial relations.

7 REFLEXIVE RELATION A Relation R in a set A is said to be reflexive relation, if (a, a) Є R, for every a Є A. Example 3 : let A be the set of all triangles drawn in a plane and R be the relation ‘is similar to’ on A, then R is a reflexive relation. REASON : every triangle is similar to itself i.e. a R a for all a Є A.

8 SYMMETRIC RELATION A Relation R in a set A is said to be symmetric relation, if (a, b) Є R implies that (a, b) Є R, for all a, b Є A. Example 4 : let A be the set of all triangles drawn in a plane and R be the relation ‘is similar to’ on A, then R is a symmetric relation. REASON : If a triangle a is similar to a triangle b, then the triangle b is also similar to the triangle a i.e. a R b implies b R a.

9 TRANSITIVE RELATION A Relation R is said to be transitive relation, if (a, b) Є R and (b, c) Є R implies that (a, c) Є R, for all a, b, c Є A. Example 5 : let A be the set of all triangles drawn in a plane and R be the relation ‘is similar to’ on A, then R is a transitive relation. REASON : if triangle a is similar to triangle b and triangle b is similar to triangle c, then evidently triangle a is similar to triangle c. i.e. a R b, b R c implies a R c.

10 EQUIVALENCE RELATION A Relation R is said to be an equivalence relation, if R is reflexive, symmetric and transitive. Example 6 : Let R be a relation on the set of all lines in a plane defined by (a, b) Є R implies that line a is parallel to line b. Show that R is an equivalence relation.

11 Solution : let A be the given set of all lines in a plane. Then, we observe the following properties. Reflexive: For each line a ЄA, we have a||a (a, a) Є R for all a Є A. So, R is reflexive. Symmetric: let a, b Є A such that (a, b) Є R. Then, (a, b) Є R, a||b b||a. (b, a) Є R. So, R is symmetric on A.

12 Transitive : let a, b, c Є A such that (a, b) Є R and (b, c) Є R. Then, (a, b) Є R and (b, c) Є R, a||b and b||c. a||c (a, c) Є R. So, R is a transitive. Hence, R being reflexive, symmetric and transitive is an equivalence relation on A.

13 Example 7 : Show that the relation R on the set A of all books in a library of a college given by R = {(x, y): x and y have the same number of pages}, is an equivalence relation. Solution : we observe the following properties of relation R. Reflexive : For any book x in a set A, we observe that x and x have same number of pages. (x, x) Є R (x, x) Є R thus,(x, x) Є R for all x Є A. thus,(x, x) Є R for all x Є A. So, R is reflexive. So, R is reflexive.

14 Symmetric : let (x, y) Є R. Then, (x, y) Є R x and y have the same number of pages y and x have the same number of pages (y, x) Є R Thus, (x, y) Є R (y, x) Є R. so, R is symmetric.

15 Transitive : let (x, y) Є R and (y, z) Є R. Then, (x, y) Є R and (y, z) Є R. x and y have the same number of pages and y and z have the same number of pages x and z have the same number of pages. (x, z) Є R. So, R is transitive. Thus, R is reflexive, symmetric and transitive. hence, R is an equivalence relation.

Example 8 : Let R be the set of all real Numbers and R be the relation on Rdefined Numbers and R be the relation on R defined by R = {(a, b):|a| ≤ b}. Show that R is neither by R = {(a, b):|a| ≤ b}. Show that R is neither reflexive nor symmetric but transitive. reflexive nor symmetric but transitive. Solution : R is not reflexive. Take a = -1, we note that (-1, -1) R, because|-1| ≤ -1 i.e. 1 ≤ -1 is wrong. R is not symmetric. 16

17 Take a = 1, b = 2, then |1| ≤ 2 i.e. 1 ≤ 2 is true (1, 2) Є R but (2, 1) R, because |2| ≤ 1 i.e. 2 ≤ 1 is wrong. R is transitive. If (a, b) Є R and (b, c) Є R then |a| ≤ b and |b| ≤ c. |a| ≤ c ( b ≤ |b| for all b Є R) (a, c) Є R R is transitive.

Example 9 : Show that the relation R on the set R of real numbers, defined as set R of real numbers, defined as R ={(a, b): a ≤ b 2 }, is neither reflexive nor symmetric nor transitive. Solution : Given R ={(a, b): a, b Є R, a ≤ b 2 }. i)R is not reflexive. take a = ½. As ½ ≤ (½) 2 i.e. ½ ≤ ¼ is false, take a = ½. As ½ ≤ (½) 2 i.e. ½ ≤ ¼ is false, (½, ½) R. Therefore, R is not reflexive. (½, ½) R. Therefore, R is not reflexive. 18

ii)R is not symmetric. take a = 1, b = 2. As 1 ≤ 2 2 i.e. 1 ≤ 4 is true, take a = 1, b = 2. As 1 ≤ 2 2 i.e. 1 ≤ 4 is true, (1, 2) Є R. But 2 ≤ 1 2 i.e.2 ≤ 1 is false, (1, 2) Є R. But 2 ≤ 1 2 i.e.2 ≤ 1 is false, (2, 1) R. Therefore, R is not symmetric. (2, 1) R. Therefore, R is not symmetric. iii)R is not transitive. take a =, b = -2 and c = ½. take a =, b = -2 and c = ½. As ≤ (-2) 2 i.e. 1/3 ≤ 4 is true, As ≤ (-2) 2 i.e. 1/3 ≤ 4 is true, (, -2) Є R. (, -2) Є R. but ≤ (½) 2 i.e. ≤ ¼ is false, but ≤ (½) 2 i.e. ≤ ¼ is false, (,½) R. (,½) R. Therefore, R is not transitive. Therefore, R is not transitive. 19

1)Let T be the set of all triangles in a plane with R a relation in T given by R = {(T 1, T 2 ): T 1 is congruent to T 2 }. Show that R is an equivalence relation. R = {(T 1, T 2 ): T 1 is congruent to T 2 }. Show that R is an equivalence relation. 2)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.

3 ) Give an example of a relation. Which is i) Symmetric but neither reflexive nor i) Symmetric but neither reflexive nor transitive. transitive. ii) Transitive but neither reflexive nor ii) Transitive but neither reflexive nor symmetric. symmetric. iii) Reflexive and symmetric but not iii) Reflexive and symmetric but not transitive. transitive. iv) Reflexive and transitive but not iv) Reflexive and transitive but not symmetric. symmetric. v) Symmetric and transitive but not v) Symmetric and transitive but not reflexive. reflexive. 21

22