If P = a , b , c , find P × P .

We know that, the cross product of two sets is defined as A × B = x , y , x ∈ A , y ∈ B . Also if n A = p and n B = q then n A × B = p q , where n A denotes the number of elements in the set A . Given P = a , b , c . ∴ P × P = a , a , a , b , a , c , b , a , b , b , b , c , c , a , c , b , c , c .

Let A = 2 , 3 and B = 5 , 6 . Write A × B . How many subsets will A × B have?

We know that, the cartesian product of two non-empty sets X and Y is defined as X × Y = x , y : x ∈ X , y ∈ Y . Given A = 2 , 3 and B = 5 , 6 . Hence A × B = 2 , 4 , 2 , 5 , 3 , 4 , 3 , 5 . Thus, the number of elements in A × B is 4 . We also know that a set with n elements will have 2 n subsets. Since, n A × B = 4 , hence A × B will have 2 4 = 16 subsets.

Let A = 1 , 3 , 6 , 9 . Let R be the relation on A defined by R = { x , y : x ∈ A , y ∈ A and x divides y } . Find R in roster form.

Given that A = 1 , 3 , 6 , 9 . Here, R = { x , y : x ∈ A , y ∈ A and x divides y } . Using the definition the relation R in roster form can be written as R = 1 , 1 , 1 , 3 , 1 , 6 , 1 , 9 , 3 , 3 , 3 , 6 , 3 , 9 , 6 , 6 , 9 , 9 .

Find the domain of the relation R given by R = { x , y : x ∈ N , y ∈ N and x + y = 5 } .

Given R = { x , y : x ∈ N , y ∈ N and x + y = 5 } . Then, the relation R in roster form is R = 1 , 4 , 2 , 3 , 3 , 2 , 4 , 1 . We know that in a relation, if x is related to y , then, the set of the values of x is the domain of the relation. Hence, the domain of R is 1 , 2 , 3 , 4 .

Find the range of the relation R given by R = { x , y : x ∈ N , y ∈ N and x 2 + y 2 ≤ 24 .

Given R = { x , y : x ∈ N , y ∈ N and x 2 + y 2 ≤ 24 . Hence R = 1 , 1 , 1 , 2 , 1 , 3 , 1 , 4 , 2 , 1 , 2 , 2 , 2 , 3 , 2 , 4 , 3 , 1 , 3 , 2 , 3 , 3 , 4 , 1 , 4 , 2 . We know that in a relation, if x is related to y , then, the set of the values of y is the range of the relation. Thus, the range of R is 1 , 2 , 3 , 4 .

Let A = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 . If R be the relation on A defined by R = { x , y : x ∈ A , y ∈ A and y = 3 x } . Find the range of R .

Given that A = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 . A relation R is defined on A as R = { x , y : x ∈ A , y ∈ A and y = 3 x } . Hence, using the given definition, we get R = 1 , 3 , 2 , 6 , 3 , 9 . We know that in a relation, if x is related to y , then, the set of the values of y is the range of the relation. Hence, the range of R is 3 , 6 , 9 .