# Logic symbol question

• Oct 4th 2011, 02:29 PM
ehpoc
Logic symbol question
I am asked to draw a venn diagram of a few sets. But next to them there is a small C and I have no clue what that small C means.

To see it in context look at question 6 here. Thanks so much for any help!!!

CS2130 Assignment 1
• Oct 4th 2011, 02:45 PM
Plato
Re: Logic symbol question
In this context $\displaystyle A^c$ is the relative complement.
That is $\displaystyle U\setminus A=\{x\in U:~x\notin A\}$
• Oct 4th 2011, 03:11 PM
ehpoc
Re: Logic symbol question
so..

Quote:

6. Make a Venn diagram illustrating the case

Ac∩Bc∩C = Ac∩B∩Cc = ∅, B ≠ C, and B ∩ C ≠ ∅.
Few questions...

This is supposed to be one single venn diagram.

This is asking me to show that...
A compliment intersect B compliment intersect C = A compliment intersect B intersect C compliment=null set?

So is also says that B does not intersect C. So if B does not intersect C and I show Ac∩Bc∩C I end up with the area of C shading in that does not intersect A, which does not equal the null set.

Ac∩B∩Cc for this I get the area of B shaded in that does not intersect A.

Here is how I got it those two.

First this means B ∩ C ≠ ∅ that B and C do not intersect.

So my venn is circle B intersecting circle A which intersects circle C.

So A compliment is everything outside of A.

B compliment is everything outside of B

Ac intersect Bc is everything outside of A and B

and Ac intersect Bc intersect C is everywhere that is both outside of A and B and in C which ends up being the area of C not intersect by A. So I am confused at how this can = the null set.
• Oct 4th 2011, 03:20 PM
Plato
Re: Logic symbol question
Here is the Venn diagram of $\displaystyle A^c\cap B^c\cap C$
• Oct 4th 2011, 03:27 PM
ehpoc
Re: Logic symbol question
Yes that is what I got!!! So how can that possibly =∅?????? I am confused!
• Oct 4th 2011, 03:33 PM
Plato
Re: Logic symbol question
Quote:

Originally Posted by ehpoc
Yes that is what I got!!! So how can that possibly =∅?????? I am confused!

Well now do the same for $\displaystyle A^c\cap B\cap C^c$.
Do you see any overlapping?
• Oct 4th 2011, 03:50 PM
ehpoc
Re: Logic symbol question
Ya but the question asked to show..

Quote:

Ac∩Bc∩C = Ac∩B∩Cc = ∅
Not

Quote:

(Ac∩Bc∩C) ∩ Ac∩B∩Cc = ∅
unless does that "=" in the middle mean intersection???
• Oct 4th 2011, 03:58 PM
Plato
Re: Logic symbol question
Quote:

Originally Posted by ehpoc
Ya but the question asked to show..

Not

unless does that "=" in the middle mean intersection???

But you are also given that $\displaystyle B\cap C=\emptyset$
• Oct 4th 2011, 04:24 PM
ehpoc
Re: Logic symbol question
Quote:

But you are also given that B\cap C=\emptyset
Which I thought meant that B and C do not intersect anywhere.

So Ac∩Bc∩C is...

Attachment 22488

and Ac∩B∩Cc is..

Attachment 22489

Neither of these diagrams are equal and neither = \emptyset