Let A = {–3, –1}, B = {1, 3) and C = {3, 5). Find:

(i) Given: $A=\{-3,-1\}$ and $B=\{1,3\}$

To find: A × B

By the definition of the Cartesian product

Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P and Q, .i.e.

$P \times Q=\{(p, q): p \in P, q \in Q\}$

Here, $A=\{-3,-1\}$ and $B=\{1,3\} .$ So,

$A \times B=\{-3,-1\} \times\{1,3\}$

$=\{(-3,1),(-3,3),(-1,1),(-1,3)\}$

(ii) Given: $C=\{3,5\}$

From part (i), we get $A \times B=\{(-3,1),(-3,3),(-1,1),(-1,3)\}$

So

$(A \times B) \times C=\{(-3,1),(-3,3),(-1,1),(-1,3)\} \times(3,5)$

$=(-3,1,3),(-3,1,5),(-3,3,3),(-3,3,5),(-1,1,3),(-1,1,5),(-1,3,3),(-1,3,5)\}$

(iii) Given: $B=\{1,3\}$ and $C=\{3,5\}$

To find: B $\times \mathrm{C}$

By the definition of the Cartesian product

Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P and Q, .i.e.

$P \times Q=\{(p, q): p \in P, q \in Q\}$

Here, $B=\{1,3\}$ and $C=\{3,5\} .$ So,

$B \times C=(1,3) \times(3,5)$

$=\{(1,3),(1,5),(3,3),(3,5)\}$

(iv) Given: $A=\{-3,-1\}$

From part (iii), we get $B \times C=\{(1,3),(1,5),(3,3),(3,5)\}$

So,

$A \times(B \times C)=\{-3,-1\} \times\{(1,3),(1,5),(3,3),(3,5)\}$

$=(-3,1,3),(-3,1,5),(-3,3,3),(-3,3,5),(-1,1,3),(-1,1,5),(-1,3,3),(-1,3,5)\}$