Can you prove the distributive laws?

[phillyolly]

Homework Statement

This is not a HW, I am reviewing the basic material for my own use.

Please prove the distributive laws.

Homework Equations

The Attempt at a Solution

 

[Raskolnikov]

Read my post about proving equivalence relations on your other thread "Proof (Real Analysis I)." The same concept of needing to prove it forwards and backwards applies here. Simply substitute in the appropriate premises and conclusions.

 

[phillyolly]

Read my post about proving equivalence relations on your other thread "Proof (Real Analysis I)." The same concept of needing to prove it forwards and backwards applies here. Simply substitute in the appropriate premises and conclusions.

Hi,
Thank you for the posts. I read you previous post on equivalence. I am a person with zero experience in Real Analysis. What you wrote made little sense to me. If you can do either of these problems, it will jump start my process of learning. Can I ask you to solve either problem step-by-step?
Again, these are not my HW questions.

 

[Raskolnikov]

If you have zero experience, then I would recommend teaching yourself or taking a quick course on boolean logic and point-element set theory first. I'll go into (almost excess) detail below to help you out here, but you should be able to get this.

Part (a):

$$( \Rightarrow )$$
Let $$x \in ( A \cap (B \cup C) ).$$
Then $$(x \in A) \wedge (x \in B \vee x \in C).$$
Thus, by the distributive law, $$(x \in A \wedge x \in B) \vee (x \in A \wedge x \in C).$$
Hence, $$x \in (A \cap B) \vee x \in (A \cap C).$$
Therefore, $$x \in ((A \cap B) \cup (A \cap C)).$$

Note: we are just halfway through the proof. We have proved the statement in the forward direction. Now we need to prove it backwards, i.e., $$x \in ((A \cap B) \cup (A \cap C)) \rightarrow x \in ( A \cap (B \cup C) ).$$

 

[phillyolly]

This was so extremely helpful. I am very thankful to you, Raskolnikov.