When will A × B be equal to B × A ?

If P and Q are two non-empty sets, then the set of all ordered pairs p , q such that p ∈ P , q ∈ Q is called the Cartesian product of P and Q , and is denoted by P × Q . So, A × B = a , b | a ∈ A , b ∈ B ... ( 1 ) and, B × A = { b , a | b ∈ B , a ∈ A } ... ( 2 ) Thus, from ( 1 ) and ( 2 ) , A × B and B × A will be equal, when either A = B or at least one of the set between A and B is an empty set.

Find A × B , A × A and B × A , if A = { 2 , - 2 , 3 } and B = { 1 , - 4 } .

Given, A = { 2 , - 2 , 3 } , B = { 1 , - 4 } The set of all points in the Cartesian plane can be viewed as the set of all ordered pairs ( x , y ) , where x , y are real numbers. In fact, R × R is the set of all points which we call as the Cartesian plane. We have to find the Cartesian product of A × B , A × A , B × A . If A and B are two non-empty sets, then the set of all ordered pairs a , b such that a ∈ A , b ∈ B is called the Cartesian product of A and B and is denoted by A × B . A × B = a , b | a ∈ A , b ∈ B , Similarly, it is the same for A × A and B × A . Hence, A × B = 2 , - 2 , 3 × 1 , - 4 = { ( 2 , 1 ) , ( 2 , - 4 ) , ( - 2 , 1 ) , ( - 2 , - 4 ) , ( 3 , 1 ) , ( 3 , - 4 ) } . A × A = 2 , - 2 , 3 × 2 , - 2 , 3 = { ( 2 , 2 ) , ( 2 , - 2 ) , ( 2 , 3 ) , ( - 2 , 2 ) , ( - 2 , - 2 ) , ( - 2 , 3 ) , ( 3 , 2 ) , ( 3 , - 2 ) , ( 3 , 3 ) } . B × A = 1 , - 4 × 2 , - 2 , 3 = { ( 1 , 2 ) , ( 1 , - 2 ) , ( 1 , 3 ) , ( - 4 , 2 ) , ( - 4 , - 2 ) , ( - 4 , 3 ) } .

Find A × B , A × A and B × A , if A = B = { p , q } .

Given, A = B = { p , q } . We have to find the Cartesian product of A × B , A × A , B × A . The set of all points in the Cartesian plane can be viewed as the set of all ordered pairs ( x , y ) , where x , y are real numbers. In fact, R × R is the set of all points, which we call as the Cartesian plane. If A and B are two non-empty sets, then the set of all ordered pairs a , b , such that a ∈ A , b ∈ B , is called the Cartesian product of A and B and is denoted by A × B . Thus, A × B = a , b | a ∈ A , b ∈ B . Similarly, it is the same for A × A and B × A . Hence, A × B = p , q × p , q = { ( p , p ) , { p , q ) , ( q , p ) , ( q , q ) } . A × A = p , q × p , q = { ( p , p ) , ( p , q ) , ( q , p ) , ( q , q ) } . B × A = p , q × p , q = { ( p , p ) , { p , q ) , ( q , p ) , ( q , q ) } .

Find A × B , A × A and B × A , if A = { m , n } ; B = ϕ .

Given, A = { m , n } and B = ϕ . We have to find the Cartesian product of A × B , A × A , B × A . The set of all points in the Cartesian plane can be viewed as the set of all ordered pairs ( x , y ) where x , y are real numbers. In fact, R × R is the set of all points, which we call as the Cartesian plane. If A and B are two non-empty sets, then the set of all ordered pairs a , b , such that a ∈ A , b ∈ B is called the Cartesian product of A and B and is denoted by A × B . Thus, A × B = a , b | a ∈ A , b ∈ B . Similarly, it is the same for A × A and B × A . But, here B is an empty set. So, the product of B by any other non-empty set will be an empty set. Hence, A × B = m , n × ϕ = . Also, A × A = m , n × m , n = { ( m , m ) , ( m , n ) , ( n , m ) , ( n , n ) } . And, B × A = ϕ × { m , n } = .