Proof for distributive laws by venn diagram:

The distributive laws are:

**AU(BnC) = (AUB)n(AUC)****An(BUC) = (AnB)U(AnC)**

We will first take the first law, AU(BnC) = (AUB)n(AUC).

To prove this first we have to take the left hand side. In that first we have to draw the diagram for BnC and after that we have to do the union with set A.

Next we have to take the right hand side. In that first we have to draw the diagrams for AUB and AUC seperately. Finally we have to find the union of these two.

__Left hand side:__

__Right hand side:__

The last diagram of both left hand side and right hand side are same.

So AU(BnC) = (AUB)n(AUC) is proved by venn diagram.

Now we will take the second law

An(BUC) = (AnB)U(AnC)

To prove this we will take the left hand side first and draw the diagram for BUC. After that we will do the intersection with the set A.

Next we have to take the right hand side and draw the diagrams for AnB and Anc. Now we will do the union of both the sets.

__Left hand side:__

__Right hand side:__

The last diagram of both left hand side and right hand side are same. So

An(BuC) = (AnB)u(AnC) is proved by Venn diagram.

Similarly we can prove De Morgan's law by venn diagram. Click the next page to view the proof of De Morgan's law