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 .
(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 A and 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.
(i)
There are two sets,
The set
The set
The Cartesian product of
The Cartesian product of two sets is,
Hence, equations (1) and (2) are not the same, so the given statement is false.
(ii)
There are two non-empty sets,
The Cartesian product of two non-empty sets should be a non-empty set.
The Cartesian product of set
Hence, the given statement is true.
(iii)
There are two sets,
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.
