To find (A × B) (A × C) and (A × B) (A × C), we first need to calculate the Cartesian products A × B and A × C.

Now we find the intersection of the two Cartesian products A × B and A × C. We look for ordered pairs that are present in both sets.

For the last part, we find the union of the two Cartesian products A × B and A × C. This set will contain all unique ordered pairs from both A × B and A × C.

Let $\mathrm{A}=\{1{,}2{,}3\}, \mathrm{B}=\{3{,}4\}$ and $\mathrm{C}=\{4{,}5{,}6\}$ . Find (i) $\mathrm{A} \times(\mathrm{B} \cap \mathrm{C})$ (ii) $(\mathrm{A} \times \mathrm{B}) \cap(\mathrm{A} \times \mathrm{C})$ (iii) $\mathrm{A} \times(\mathrm{B} \cup \mathrm{C})$ (iv) $(\mathrm{A} \times \mathrm{B}) \cup(\mathrm{A} \times \mathrm{C})$

Answer: (i) $\mathrm{A} \times (\mathrm{B} \cap \mathrm{C}) = \{(1{,}4), (2{,}4), (3{,}4)\}$ (ii) $(\mathrm{A} \times \mathrm{B}) \cap (\mathrm{A} \times \mathrm{C}) = \{(1{,}4), (2{,}4), (3{,}4)\}$ (iii) $\mathrm{A} \times (\mathrm{B} \cup \mathrm{C}) = \{(1{,}3), (1{,}4), (1{,}5), (1{,}6), (2{,}3), (2{,}4), (2{,}5), (2{,}6), (3{,}3), (3{,}4), (3{,}5), (3{,}6)\}$ (iv) $(\mathrm{A} \times \mathrm{B}) \cup (\mathrm{A} \times \mathrm{C}) = \{(1{,}3), (1{,}4), (1{,}5), (1{,}6), (2{,}3), (2{,}4), (2{,}5), (2{,}6), (3{,}3), (3{,}4), (3{,}5), (3{,}6)\}$

Step-by-step solution

Step 1: Calculate B intersection C

First, we find the intersection of sets B and C. The intersection of two sets contains all elements that are common to both sets. In this case, the only common element between B and C is 4.

Step 2: Calculate A cross (B intersection C)

Now we calculate the Cartesian product of set A with the result of the intersection, $\mathrm{B} \cap \mathrm{C}$ . The Cartesian product of two sets creates ordered pairs where the first element comes from the first set and the second element comes from the second set.

Step 3: Calculate A cross B and A cross C

To find $(\mathrm{A} \times \mathrm{B}) \cap (\mathrm{A} \times \mathrm{C})$ and $(\mathrm{A} \times \mathrm{B}) \cup (\mathrm{A} \times \mathrm{C})$ , we first need to calculate the Cartesian products $\mathrm{A} \times \mathrm{B}$ and $\mathrm{A} \times \mathrm{C}$ .

Step 4: Calculate (A cross B) intersection (A cross C)

Now we find the intersection of the two Cartesian products $\mathrm{A} \times \mathrm{B}$ and $\mathrm{A} \times \mathrm{C}$ . We look for ordered pairs that are present in both sets.

Step 5: Calculate B union C

For the third part, we first find the union of sets B and C. The union of two sets contains all unique elements from both sets.

Step 6: Calculate A cross (B union C)

Finally, we calculate the Cartesian product of set A with the union of B and C. This involves forming ordered pairs with each element of A and each element of $(\mathrm{B} \cup \mathrm{C})$ .

Step 7: Calculate (A cross B) union (A cross C)

For the last part, we find the union of the two Cartesian products $\mathrm{A} \times \mathrm{B}$ and $\mathrm{A} \times \mathrm{C}$ . This set will contain all unique ordered pairs from both $\mathrm{A} \times \mathrm{B}$ and $\mathrm{A} \times \mathrm{C}$ .

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