Answer
Proof: Suppose A, B, and C are any sets and A ⊆ B. Let x ϵ A ⋃ C. By definition of union, x ϵ A or x ϵ C. But since x ϵ A ⋃ C and x ϵ A, then x ϵ B. Therefore, x ϵ B or x ϵ C. By definition of intersection, x ϵ B⋃C. In conclusion, A ⋃ C ⊆ B ⋃ C.
Work Step by Step
Proof: Suppose A, B, and C are any sets and A ⊆ B. Let x ϵ A ⋃ C. By definition of union, x ϵ A or x ϵ C. But since x ϵ A ⋃ C and x ϵ A, then x ϵ B. Therefore, x ϵ B or x ϵ C. By definition of intersection, x ϵ B⋃C. In conclusion, A ⋃ C ⊆ B ⋃ C.
