Answer to Question #128903 in Abstract Algebra for Isaac Kusi

Question

 If A⊂B, prove that (A⋃C)⊂(B⋃C) for any set C.

Answer 1

$$Solution$$

If $A⊂B$ , prove that $(A⋃C)⊂(B⋃C)$ for any set $C$ .

Let $A⊂B$ , take $x\in (A⋃C)$ then $x\in A$ or $x\in C$ and thus $x\in B$

$\implies$ $x\in A$ , we can conclude that $(A⋃C)⊂A$

If $x\in A$ , then simply by definition, $x\in (A⋃C)$

$\therefore A⊂(A⋃C)$ and hence $\therefore A⊂(A⋃C)$

$\implies$ $x\in B$ we can conclude that$(B⋃C)⊂B$

If $x\in B$ , then by definition, $x\in (B⋃C)$ $\iff B⊂(B⋃C)$ $\implies \therefore B⊂(B⋃C)$

Hence the proof $(A⋃C)⊂(B⋃C)$