Question

Prove that: (i) (A ∪ B) × C = (A × C) ∪ (B × C) (ii) (A ∩ B) × C = (A × C) ∩ (B×C)

Answer 1

(i) (A ∪ B) × C = (A × C) ∪ (B × C)
Let (a, b) be an arbitrary element of (A ∪ B) × C.
Thus, we have:
$$\begin{gathered} (a,b)\in (A\cup B)\times C \\ \Rightarrow a\in (A\cup B)andb\in C \\ \Rightarrow (a\in Aora\in B)andb\in C \\ \Rightarrow (a\in Aandb\in C)or(a\in Bandb\in C) \\ \Rightarrow (a,b)\in (A\times C)or(a,b)\in (B\times C) \\ \Rightarrow (a,b)\in (A\times C)\cup (B\times C) \\ \therefore (A\cup B)\times C\subseteq (A\times C)\cup (B\times C)...\left(\mathrm{i}\right) \end{gathered}$$

Again, let (x, y) be an arbitrary element of (A × C) ∪ (B × C).
Thus, we have:
$$\begin{gathered} (x,y)\in (A\times C)\cup (B\times C) \\ \Rightarrow (x,y)\in (A\times C)or(x,y)\in (B\times C) \\ \Rightarrow (x\in A\&y\in C)or(x\in B\&y\in C) \\ \Rightarrow (x\in Aorx\in B)ory\in C \\ \Rightarrow (x\in A\cup B)\&y\in C \\ \Rightarrow (x,y)\in (A\cup B)\times C \\ \therefore (A\times C)\cup (B\times C)\subseteq (A\cup B)\times C...\left(\mathrm{ii}\right) \end{gathered}$$

From (i) and (ii), we get:
(A ∪ B) × C = (A × C) ∪ (B × C)

(ii) (A ∩ B) × C = (A × C) ∩ (B×C)
Let (a, b) be an arbitrary element of (A ∩ B) × C.
Thus, we have:
$$\begin{gathered} (a,b)\in (A\cap B)\times C \\ \Rightarrow a\in (A\cap B)\&b\in C \\ \Rightarrow (a\in A\&a\in B)\&b\in C \\ \Rightarrow (a\in A\&b\in C)\&(a\in B\&b\in C) \\ \Rightarrow (a,b)\in (A\times C)\&(a,b)\in (B\times C) \\ \Rightarrow (a,b)\in (A\times C)\cap (B\times C) \\ \therefore (A\cap B)\times C\subseteq (A\times C)\cap (B\times C)...\left(\mathrm{iii}\right) \end{gathered}$$

Again, let (x, y) be an arbitrary element of (A × C) ∩ (B × C).
Thus, we have:
$$\begin{gathered} (x,y)\in (A\times C)\cap (B\times C) \\ \Rightarrow (x,y)\in (A\times C)\&(x,y)\in (B\times C) \\ \Rightarrow (x\in A\&y\in C)\&(x\in B\&y\in C) \\ \Rightarrow (x\in A\&x\in B)\&y\in C \\ \Rightarrow x\in (A\cap B)\&y\in C \\ \Rightarrow (x,y)\in (A\cap B)\times C \\ \therefore (A\times C)\cap (B\times C)\subseteq (A\cap B)\times C...\left(\mathrm{iv}\right) \end{gathered}$$

From (iii) and (iv), we get:
(A ∩ B) × C = (A × C) ∩ (B × C)