Expert: Paul Klarreich - 10/17/2008

Question
QUESTION:  Given U = {18. 19. 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, A = {21, 23, 25, 27, 29}, and
   B = {26, 27, 28, 29}.  Find  A′ U B′.  



ANSWER: Questioner:   jalisa
Category:  Advanced Math
Private:  No
 
Subject:  Math
Question:  Given U = {18. 19. 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, A = {21, 23, 25, 27, 29}, and
  B = {26, 27, 28, 29}.  Find  A′ U B′.  
.........................................
Hi, Jalisa,

You (we) have to be careful with notation.  I assume your first U is that set, and the second U means UNION.  I think we had better not do that; we will write  A' union B' -- they aren't charging us by the letter.  Actually, it is common to write  A + B for the union, and AB for the intersection.

Now A' is the complement of A, and it means:

A' = { x | x /in A }

[Complement of A equals the set of all x such that x is not in A, but in the current universe.]

The 'universe' does not mean the stuff you see in the sky at night, but some specified or presumed set.  [In those Venn diagrams, it is marked by the enclosing rectangle.]

OK,  A = {21, 23, 25, 27, 29}, and U = {18. 19. 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, as you said, so:

What things are in U but not in A?

A' = {18, 19, 20, 22, 24, 26, 28}

and: If B = {26, 27, 28, 29}, then

B' = {18, 19, 20, 21, 22, 23, 24, 25 }

Now the union is the set of things in AT LEAST ONE of A' and B'.   I like to use those words -- it makes things more precise.

A' union B' = {18, 19, 20, 22, 24, 26, 28 PLUS 18, 19, 20, 21, 22, 23, 24, 25  deleting duplicate entries }

A' union B' = {18, 19, 20, 22, 24, 26, 28, 21, 23, 25 }

which is the answer, unless you have to put them in order. (which normally, you do only if your teacher is marking it and you don't want to drive him crazy.)



---------- FOLLOW-UP ----------

QUESTION:  Given  U = {l, m, n, o, p, q, r, s, t, u, v, w}, A = {l, m, n, o, p, q}, B = {n, o, r, s, v, w}, and C = {l, m, p, q, r, t}, find (A′ U C′) ∩ B.

Answer
Questioner:   jalisa
Category:  Advanced Math
Private:  No
 
Subject:  Set operations - basic
.........................................
Hi, Jalisa,

Given  U = {l, m, n, o, p, q, r, s, t, u, v, w}, A = {l, m, n, o, p, q}, B = {n, o, r, s, v, w}, and C = {l, m, p, q, r, t}, find (A′ + C′) ∩ B.
 
Now that we understand the notation, (I see you found out how to make the 'intersection' symbol -- some day you must show me how.) this shouldn't cause any problem:

1. Compute A' as before.
2. Compute C' as before.
3. Find their union as before.
4. FINALLY, (because of the parentheses) find the intersection of that set with B. (not B', because it does not say B')  Apply the definition:  The intersection of X and Y, written X ∩ Y, is the set of things EACH OF WHICH belongs to both sets.

My calculations give this Intersection as { l, m, p, q, r, t }.  Check it out.  

Now I wouldn't be a good teacher if I didn't give you a followup exercise.  

For the above sets, for practice, confirm the following identities: That means to just check them out for these sets -- confirm that the left side and right side of each come out the same.

A' + B' = (A ∩ B)'
A' ∩ B' = (A U B)'

They have names -- they are called DeMorgan's Laws.