Question

Prove that for any sets $\mathit{A}\mathbf{,}\mathit{B}\mathbf{,}\mathbf{}\mathit{C}\mathbf{:}$

$\mathit{A}\mathbf{\times }\mathbf{(}\mathit{B}\mathbf{\cup }\mathit{C}\mathbf{)}\mathbf{=}\mathbf{(}\mathit{A}\mathbf{\times }\mathit{B}\mathbf{)}\mathbf{\cup }\mathbf{(}\mathit{A}\mathbf{\times }\mathit{C}\mathbf{)}$

Short answer

Thus, it is proved that $(A\times B)\cup (A\times C)=A\times (B\cup C).$.

Step-by-step solution

Showing that A×(B∪C)⊆(A×B)∪(A×C)

Let (x,y) be any element of $A\times (B\cup C)$. Then,

$$\begin{gathered} \left(x,y\right)\in A\times \left(B\cup C\right) \\ \Rightarrow X\in Aandy\in \left(B\cup C\right) \\ \Rightarrow X\in Aand\left(y\in Bory\in C\right) \\ \Rightarrow \left(X\in Aandy\in B\right)or\left(x\in Aory\in C\right) \\ \Rightarrow \left(x,y\right)\in \left(A\times B\right)\cup \left(A\times C\right) \\ A\times \left(B\cup C\right)\underline{\subset }\left(A\times B\right)\cup \left(A\times C\right)....\left(1\right) \end{gathered}$$

Showing that (A×B)∪(A×C)⊆A×(B∪C)

Again, let$(u,v)$ be any element of$(A\times B)\cup (A\times C).$ .

Then, solve as follows:

$$\begin{gathered} \left(u,v\right)\in \left(A\times B\right)\cup \left(A\times C\right) \\ \Rightarrow \left(u,v\right)\in \left(A\times B\right)or\left(u,v\right)\in \left(A\times C\right) \\ \Rightarrow \left(u\in Aandv\in B\right)or\left(u\in Aandv\in c\right) \\ \Rightarrow u\in Aandv\in \left(B\cup C\right) \\ \Rightarrow \left(u,v\right)\in A\times \left(B\cup C\right) \\ \left(A\times B\right)\cup \left(A\times C\right)\underline{\cup }A\times \left(B\cup C\right) \end{gathered}$$

From (1) and (2) we get,

$(A\times B)\cup (A\times C)=A\times (B\cup C)$

Hence, it is proved that $(A\times B)\cup (A\times C)=A\times (B\cup C)$.