**Question:**

If A = {2, 3, 5} and B = {5, 7}, find:

(i) $A \times B$

(ii) $\mathbf{B} \times \mathbf{A}$

(iii) $\mathbf{A} \times \mathbf{A}$

(iv) $\mathrm{B} \times \mathrm{B}$

**Solution:**

(i) Given: A = {2, 3, 5} and B = {5, 7}

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, $A=\{2,3,5\}$ and $B=\{5,7\}$. So,

$A \times B=(2,3,5) \times(5,7)$

$=\{(2,5),(3,5),(5,5),(2,7),(3,7),(5,7)\}$

(ii) Given: $A=\{2,3,5\}$ and $B=\{5,7\}$

To find: B × A

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, $A=\{2,3,5\}$ and $B=\{5,7\}$. So,

$B \times A=(5,7) \times(2,3,5)$

$=\{(5,2),(5,3),(5,5),(7,2),(7,3),(7,5)\}$

(iii) Given: $A=\{2,3,5\}$ and $B=\{2,3,5\}$

To find: A × A

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, $A=\{2,3,5\}$ and $A=\{2,3,5\} .$ So,

$A \times A=(2,3,5) \times(2,3,5)$ $=\{(2,2),(2,3),(2,5),(3,2),(3,3),(3,5),(5,2),(5,3),(5,5)\}$

(iv) Given: B = {5, 7}

To find: B × B

By the definition of the Cartesian product,

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

Here, $B=\{5,7\}$ and $B=\{5,7\} .$ So,

$B \times B=(5,7) \times(5,7)$

$=\{(5,5),(5,7),(7,5),(7,7)\}$