A Mathematic question(about Sets)!?

Question

A crazy question!Can anyone help me to solve this question?

By using De Morgan's Law and other properties of set algebra,prove that

(A'nBnC) U {A' U (BnC')}' = (AnB') U (BnC)

U=Union
n=Intersection
' =Complement

Few laws on Sets:
1)Commutative Law
2)Associative Law
3)Distributive Law
4)De Morgan's Law:
(i)(AUB)' = A' n B'
(ii)(A n B)' = A'UB'

Answer 1

Since it looks like you are doing homework, I'll point you at some references rather than doing the math for you.

This page has some Venn diagrams that might help you visualize the operations that you need to perform.
http://www.math.ubc.ca/~feldman/m302/deMorgan.pdf#search='demorgan%27s%20law'

This page has examples of proofs using DeMorgan's rules of set operations:
http://www.cs.virginia.edu/~jck/cs202/slides/more.on.sets.pdf#search='sets%20demorgan%27s%20law'

A little more detailed but also containing some example proofs:
http://cms.dt.uh.edu/Faculty/DelavinaE/f04/2305/Ch2sec1and2SetsKoshy.pdf#search='sets%20demorgan%27s%20law'\

One more:
http://www.cs.odu.edu/~toida/nerzic/content/set/frame-set_op_prop.html

(You'll need a PDF viewer to view them. You can get one at:
http://www.adobe.com/products/acrobat/readstep2.html )

Answer 2

I'm not quite sure, but I think I have this.

(A'nBnC) U (A' U (BnC'))' = (AnB') U (BnC)

I decompose the left side, as it is easier. Breaking down the second of the two parts yields:

(A' U (BnC'))'
((A' U B) n (A' U C))'
(A' U B)' U (A' U C)'
(A n B') U (A n C')

The left side is a little trickier. I had to improvise a rule here, but it should work:

(A' n B n C)
(A' n C) n (B n C)

Those two should be equivalent statements. Combining these two terms leads to:

(A' n C) n (B n C) U (A n B') U (A n C')

And I believe that they can be rearranged so that

(A' n C) n (A n C') U (A n B') U (B n C)

which leads to

(A n B') U (B n C)

Answer 3

BnC is a common set in the venn diagram..

Answer 4

just apply all the laws to simply the left side

Answer 5

whoa. r u in AP Calculus? dayum i didnt pass Pre-Calculus during my 10th grade so "F math!"
whose De Morgan?

Answer 6

{(a' n b n c ) u (a' u (b n c'))}'
=(a' n b n c)' n (a' u (b n c') )'
=(a' n b n c)' n (a n (b n c')')
=(a' n b n c)' n (a n (b u c) )
=... ?.?


i'm sorry i never learn how to simplify (a n b n c )' (3 variables)
it never been taught at school
if you can tell me how, maybe i can solve this question

Answer 7

I don't know the answer, but I'm certainly impressed that you know the question.