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I was trying to prove Distributive law ie A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ). I tried to prove it in traditional way:

Let X ∈ A ∩ ( B ∪ C ). Then X ∈ A and X ∈ B ∪ C. Then X ∈ A and X ∈ B $or$ X ∈ C. [ Here $or$ is used in inclusive way]

After this I am not able to solve it. What can I infer from X ∈ B $or$ X ∈ C? Do I consider X belongs only to B or X belongs only to C or X belongs to both B and C?

Andrés E. Caicedo
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  • Since you know $X\in B$ or $X\in C$, you can just consider the two cases (which happen to be almost identical here). – 79037662 Oct 16 '19 at 14:19

2 Answers2

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Hint. If $X\in A$ and $X\in B$, then $X\in A\cap B$. Otherwise, then $X\in A$ and $X\in C$, so $X\in A\cap C$. Either way, $X\in(A\cap B)\cup(A\cap C)$. But this doesn't complete the proof, since we've only shown inclusion in one direction. Can you do the other?

YiFan Tey
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We have to prove both inclusions $$ A\cap ( B\cup C)\subseteq ( A\cap B) \cup ( A\cap C)\quad\mbox{ and } \quad A\cap ( B\cup C)\supseteq ( A\cap B) \cup ( A\cap C). $$ Let's prove the first. The second is similarly proven and is an exercise for you. We have each of the following statements implying the next.

  • $x\in A\cap ( B\cup C)$
  • $x\in A$ and $x\in B\cup C$
  • $x\in A$ and $x\in B$ or $ x\in C$
  • $x\in A$ or $x\in B$ and $x\in A$ or $ x\in C$
  • $x\in A\cup B$ and $x\in A\cup C$
  • $x\in (A\cup B)\cap (A\cup B)$

So we have to $A\cap ( B\cup C)\subseteq (A\cup B)\cap (A\cup B)$

John Strood
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Elias Costa
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  • @Bram28 It is true. But I wanted to let the questioner realize that. – Elias Costa Oct 16 '19 at 14:35
  • Ah! So I screwed up a pedagogical moment! :( OK, I'll erase my comment – Bram28 Oct 16 '19 at 14:52
  • In step 4 you have wrote: x∈ A or x∈ B and x∈ A or x ∈ C. But I think this should have been: X ∈ A and X ∈ B or X ∈ A and X ∈ C. Is it right? If not please give reason why it is wrong. –  Oct 17 '19 at 00:54