Question

State whether each of the following statement are true or false. If the statement is false, rewrite the given statement correctly. (i) If P = { m , n } and Q = { n , m }, then P × Q = {( m , n ), ( n , m )}. (ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs ( x , y ) such that x ∈ A and y ∈ B. (iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ ) = Φ .

Answer 1

(i)

There are two sets, P and Q.

The set P is,

P={ m,n }

The set Q is,

Q={ n,m }

The Cartesian product of P×Q is given as,

P×Q={ ( m,n ),( n,m ) }(1)

The Cartesian product of two sets is,

P×Q={ ( m,n ),( m,m ),( n,n ),( n,m ) }(2)

Hence, equations (1) and (2) are not the same, so the given statement is false.

(ii)

There are two non-empty sets, A and B.

The Cartesian product of two non-empty sets should be a non-empty set.

The Cartesian product of set Aand set B is,

A×B={ ( x,y );x∈A,y∈B }

Hence, the given statement is true.

(iii)

There are two sets, A and B.

A={ 1,2 } and B={ 3,4 }

B∩ϕ={ 3,4 }∩ϕ =ϕ

A×( B∩ϕ )={ 1,​2 }×ϕ =ϕ

In the Cartesian product of two sets, if one set is a null set, then the Cartesian product

of the two sets should be a null set.

Hence, the given statement is true.