# sets

• Jun 18th 2012, 11:03 PM
serhanbener
sets
A\(B ∩ C) ⊆(A\B) ∩ (A\C)
How can I show this on a diagram?

How can I draw a diagram for this set?
• Jun 19th 2012, 02:37 AM
emakarov
Re: sets
Quote:

Originally Posted by serhanbener
How can I draw a diagram for this set?

You have two sets, not one: A \ (B ∩ C) and (A \ B) ∩ (A \ C).

Quote:

Originally Posted by serhanbener
How can I show this on a diagram?

I assume you mean Venn diagrams. Do you know how to show anything on a Venn diagram, such as the set A or B ∩ C? If not, then you need to read a textbook. If yes, then what exactly is your difficulty in drawing these slightly more complicated sets?

I believe the inclusion you wrote is false in general. It would be true and, in fact, in would be an equality if ∩ is replaced by ∪ in the right-hand side.
• Jun 19th 2012, 03:38 AM
serhanbener
Re: sets
Yes I know how to show anything on a Venn diagram. But I can't show the example above.
• Jun 19th 2012, 04:38 AM
emakarov
Re: sets
Attachment 24104

(The picture is clickable.)

In the left picture, B ∩ C is blue and A is red. The part of A that is just red, i.e., does not include the central purple region, is A \ (B ∩ C).

In the right picture, A \ B is red and A \ C is blue. The purple intersection is (A \ B) ∩ (A \ C).

You can see that the purple part in the right picture is a subset of a purely red part of the left picture, i.e., (A \ B) ∩ (A \ C) ⊆ A \ (B ∩ C). Also, the purely red part of the left picture equals the painted part of the right picture, i.e., (A \ B) ∪ (A \ C) = A \ (B ∩ C).
• Jun 21st 2012, 01:19 AM
serhanbener
Re: sets
Many Thanks. I think there is a problem with the question."A\(B ∩ C) ⊆(A\B) ∩ (A\C)" seems to be wrong.