Set Theory Expression Simplification: {[(A∪B) ∩ C]' ∪ B'}' ∪ C

[IrishGuy99]

Set Theory Expression Simplification: {[(A∪B) ∩ C]' ∪ B'}' ∪ C

Hello,
I have been trying to solve a problem related to sets theory expression. I have been asked to simplify it using the laws associated with sets i.e associative law, deMorgans, involution etc.) I have tried many methods but the lecturer didnt really explain how to go about simplifying it.

Here is the problem I hope you can visualise it well enough:

Let X be a universal set and A, B, C be three subsets of X.

Simplify:[SIZE=7]
[SIZE=3]{[(A[/SIZE][/SIZE][SIZE=3][COLOR=#000000][FONT=Helvetica]∪B) ∩ C]' [/FONT][/COLOR][COLOR=#000000][FONT=Helvetica]∪ B'}' [/FONT][/COLOR][/SIZE][COLOR=#000000][FONT=Helvetica][SIZE=3]∪ C 
[/SIZE][SIZE=3]
where D' denotes the complement of any subset D of the universal set X.[/SIZE]
[/FONT][/COLOR]

I have broken down the first part by using associative laws and multiplied in the complement. I then cancelled the complement of B' using involution law where (A')' = A. But then I get to a sum that I can''t seem to get further past and it doesn't really make sense. Pointers would be really appreciated as I have tried every method I can possibly think of. I am obviously doing something wrong. But as I have said, we haven't been shown how to simplify it in class.

This is what I have tried so far to no success:

{[(C

∩ A) ∪ (C∩ B)]' ∪ B'}' ∪ C​

{[(C

∩ A)' ∪ (C ∩ B)'] ∪ B} ∪ C​

{[(C'

∪ A') ∪ (C' ∪ B')] ∪ B} ∪ C​

{[C'

∪ (A' ∪ B')] ∪ B} ∪ C​

And that is where I get stuck ( formatting keeps messing up my answer sorry)


Thanks,
IrishGuy99

 

[Dr.Peterson]

Hello,
I have been trying to solve a problem related to sets theory expression. I have been asked to simplify it using the laws associated with sets i.e associative law, deMorgans, involution etc.) I have tried many methods but the lecturer didnt really explain how to go about simplifying it.

Here is the problem I hope you can visualise it well enough:

[SIZE=3][FONT=courier new]Let X be a universal set and A, B, C be three subsets of X.

Simplify:
{[(A[COLOR=#000000]∪B) ∩ C]' ∪ B'}' [/COLOR][/FONT][/SIZE][COLOR=#000000][FONT=Helvetica][SIZE=3][FONT=courier new]∪ C 

where D' denotes the complement of any subset D of the universal set X.[/FONT][/SIZE][/FONT][/COLOR]

I have broken down the first part by using associative laws and multiplied in the complement. I then cancelled the complement of B' using involution law where (A')' = A. But then I get to a sum that I can''t seem to get further past and it doesn't really make sense. Pointers would be really appreciated as I have tried every method I can possibly think of. I am obviously doing something wrong. But as I have said, we haven't been shown how to simplify it in class.

Thanks,
IrishGuy99

It will help a lot if you show the actual work, rather than just describe it. As it is, we can't be sure what you did, and whether you did it correctly. Once we see the details, we'll probably have a lot to say.