# Thread: Prove By Distributive Law

1. ## Prove By Distributive Law

Question, Prove by distributive law :

$a_{1} \cap ( a_2 \cup ... \cup a_n ) = ( a_1 \cap a_2 ) \cup ... \cup ( a_1 \cap a_n )$ is true for all n>= 3

any one got a clue on this problem? I am lost....

2. Originally Posted by oldguy
Question, Prove by distributive law :

$a_{1} \cap ( a_2 \cup ... \cup a_n ) = ( a_1 \cap a_2 ) \cup ... \cup ( a_1 \cap a_n )$ is true for all n>= 3

any one got a clue on this problem? I am lost....
since you asked for a clue...use induction. the claim for n = 1 is by definition, use that to prove the claim true for n = 2, then use induction to show it's true for n >= 3

3. Thanks for the clue. I will try your suggestion.

4. Does any one know how to get started with this proble as Jhevon suggested?

5. Originally Posted by oldguy
Question, Prove by distributive law :

$a_{1} \cap ( a_2 \cup ... \cup a_n ) = ( a_1 \cap a_2 ) \cup ... \cup ( a_1 \cap a_n )$ is true for all n>= 3

any one got a clue on this problem? I am lost....
Well, you know that
$a_1 \cap ( a_2 \cup a_3) = (a_1 \cap a_2) \cup (a_1 \cap a_3)$

So now consider the case for
$a_1 \cap ( a_2 \cup a_3 \cup a_4)$

Define a new set $a_5 = a_3 \cup a_4$.

What is $a_1 \cap ( a_2 \cup a_5)$?

-Dan