Expert: Paul Klarreich - 2/6/2007

Question
-Concepts of Math

i have 2 questions:

1)Let A and B be subsets of a set X. Prove the de Morgan laws
(A U B)complement = A complement intersection B complement and (A intersection B)complement = A complement U B complement

2)If A and B are subsets of a set X, prove that A - B = A intersection B complamint.

i've proved that A - B is in A intersection B complamint but i don't know how to prove A intersection B complamint is in A- B  

Answer
Questioner:   Linda
Category:  Advanced Math

 
Subject:  Proofs
Question:  -Concepts of Math

I have 2 questions:

1)Let A and B be subsets of a set X. Prove the de Morgan laws
(A U B)complement = A complement intersection B complement and (A intersection

B)complement = A complement U B complement

2)If A and B are subsets of a set X, prove that A - B = A intersection B

complamint.

i've proved that A - B is in A intersection B complamint but i don't know how to prove A intersection B complamint is in A- B
...............................................
Hi, Linda,

I will use the following notation to save typing:

A' is the complement of A
AB is the intersection.
A+B is the union.
x in A  means x is an element of A
x /in A means x is not an element of A

1) To prove the laws:  

(A + B)' = A' B'

Suppose  x in (A + B).  Then  x belongs to at least one of A and B.  Then if x in (A + B)', then x DOES NOT belong to at least one of A and B.  Then it cannot belong to either one, so  x /in A  AND x /in B.  Therefore  x in A'  and x in B'.  Since x in both A' and B',  x in A'B'.
.............
(AB)' = A' + B'

Suppose  x in AB.  Then  x in A and x in B.  Then if  x in (AB)', either   x /in A  or  x /in B.  That means either x in A' or x in B', meaning  x in at least one of A' and B', which means x in A' + B'.

Now a suggestion, which your teacher might like.  Construct a Venn Diagram for the sets A and B.  (Make a rectangle that you label U, for the universe, put two overlapping circles in it, that you label A and B.)  Now there are four regions, which you number 1,2,3,4, in any order.  Suppose we label them as:

1 = the region of U that is outside both A and B.
3 = the overlap region of A and B. (obviously it's AB)
2 = the part of A not in B.
4 = the part of B not in A.

Now you just compute the left and right sides of your 'laws' and find that they come out to the same sets of regions.

2)If A and B are subsets of a set X, prove that A - B = A intersection B complamint.

>> No, a complamint is an after-dinner candy. That's COMPLEMENT.

If  A - B means "the part of A that is not in B", then you can write the definition:

A - B = { x |  x in A, but  x /in B }, you practically have your proof right there :  x in A but x /in B means  x in A and x in B', which means, in turn,  x in AB'

[Note: Mathematically, the word 'but' is identical to the word 'and'.  In elegant usage, we write 'but' when the clause after it contains a 'not' somewhere.  But the meaning is identical.]