Construct the composition table/multiplication table for the binary operation * defined on 0 , 1 , 2 , 3 , 4 by a * b = a × b (mod 5 ). Find the identity element if any. Also find the inverse elements of 2 and 4 .

Given ' * ' on 0 , 1 , 2 , 3 , 4 defined by a * b = a × b m o d 5 Consider the composition table, × 5 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 In the given composition table, all the elements are in the set 0 , 1 , 2 , 3 , 4 . Let a , b ∈ 0 , 1 , 2 , 3 , 4 Then, a × b m o d 5 ∈ 0 , 1 , 2 , 3 , 4 ⇒ a * b ∈ 0 , 1 , 2 , 3 , 4 Thus, * is a binary operation on 0 , 1 , 2 , 3 , 4 . From the table we can observe that elements of second row as same as the top-most row. So, 1 ∈ 0 , 1 , 2 , 3 , 4 is the identity element with respect to × 5 . Now we have to find the inverse of 2 and 4 . 2 × 5 3 = 1 and 4 × 5 4 = 16 So, the inverse of 2 is 3 and the inverse of 4 is 4 .

Construct the composition table/multiplication table for the binary operation * defined on 0 , 1 , 2 , 3 , 4 by a * b = a × b (mod 5 ). Find the identity element if any. Also find the inverse elements of 2 and 4 .

Given ' * ' on 0 , 1 , 2 , 3 , 4 defined by a * b = a × b m o d 5 Consider the composition table, × 5 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 In the given composition table, all the elements are in the set 0 , 1 , 2 , 3 , 4 . Let a , b ∈ 0 , 1 , 2 , 3 , 4 Then, a × b m o d 5 ∈ 0 , 1 , 2 , 3 , 4 ⇒ a * b ∈ 0 , 1 , 2 , 3 , 4 Thus, * is a binary operation on 0 , 1 , 2 , 3 , 4 . From the table we can observe that elements of second row as same as the top-most row. So, 1 ∈ 0 , 1 , 2 , 3 , 4 is the identity element with respect to × 5 . Now we have to find the inverse of 2 and 4 . 2 × 5 3 = 1 and 4 × 5 4 = 16 So, the inverse of 2 is 3 and the inverse of 4 is 4 .

If A = a , b , c , d then the relation R = a , a , b , b , c , c , d , d . Mention the types of relation R on A

Given A = a , b , c , d and relation R = a , a , b , b , c , c , d , d Reflexive : ( a , a ) ∈ R ∀ a ∈ A , where a is the element, A is the set and R is the relation. Here, the given relation contains the reflexive ordered pair for all the elements present in the set A Therefore, the relation R is reflexive. Symmetric: Let A be a set in which the relation R defined. Then R is said to be a symmetric relation, if ( a , b ) ∈ R ⇒ ( b , a ) ∈ R , for all a , b ∈ A . All the ordered pairs are symmetric Therefore, R is symmetric Transitive: Let A be a set in which the relation R defined. Then R is said to be transitive relation, if ( a , b ) ∈ R ( b , c ) ∈ R ⇒ ( a , c ) ∈ R , ∀ a , b , c ∈ A Therefore, given R is transitive relation. Hence, the given relation is equivalence relation.