# Discrete Math - Proving Distributive Laws for Sets by induction

I'm working on doing a proof by induction on this question:

Use induction to prove that if $$X_1, . . . , X_n$$ and $$X$$ are sets, then $$X∩(X_1∪X_2∪· · ·∪X_n) = (X∩X_1)∪(X∩X_2)∪· · ·∪(X∩X_n)$$.

I've shown the basis case:

$$X∩X_1 = X∩X_1$$

But I'm having difficulty proving the $$n + 1$$ case. This is my work so far:

Assume $$X∩(X_1∪X_2∪· · ·∪X_n) = (X∩X_1)∪(X∩X_2)∪· · ·∪(X∩X_n)$$

Show $$X∩(X_1∪X_2∪· · ·∪X_n∪X_{n+1}) = (X∩X_1)∪(X∩X_2)∪· · ·∪(X∩X_n) = (X∩X_1)∪(X∩X_2)∪· · ·∪(X∩X_n)∪(X∩X_{n+1})$$

I'm not sure what I'm allowed to do from here. Where can I use the induction hypothesis?

• Please use MathJax to format your question correctly – Ankit Kumar Jan 11 at 4:57

Tip show: $$X\cap(Y\cup X_{n+1})=(X\cup Y)\cap(X\cup X_{n+1})$$ where $$Y=X_1\cup\cdots\cup X_n$$