§2.3: Venn Diagrams and Set Operations
I.
MGF 1106-Peace
Universal Set
Def: The universal set, symbolized by U, is a set that contains ALL elements being considered in the problem.
We normally use a square or rectangle to describe the universal set.
U The diagram to the left is called a Venn diagram, which gives us a visual display
of the relationship between sets. It is named for John Venn.
Ex. You are doing a study on women aged 18-49 and their smoking habits. The universal set for your particular
problem would be all the women aged 18-49 that you interviewed.
To represent subsets of the universal set within Venn diagram, we normally use circles that are within the rectangle.
Ex. Continuing our previous example. Let’s say you ask the women in your survey whether or not they have EVER
smoked. Some will say yes and some will say no. Specifically you are interested in the ones who do smoke. You can
represent the set of 18-49 year old women who smoke (in your survey) as:
U The women who smoke is represented by the set A in the Venn diagram. Also,
notice that the circle A is totally within U, indicating that A is a subset of U. The
region inside the rectangle and inside the circle A are those women who smoked.
The region inside U, but outside A are those women who never smoked.
A Ex. Now, let’s say you decide to ask a second question about the women’s drinking habits. Specifically you ask, “Do
you drink alcohol?” Those surveyed either say yes or no. Now a second circle will be introduced into our Venn
diagram. Specifically, we will denote B as those surveyed who do consume alcohol.
U A I B
II III IV Now we have two overlapping circles in the diagram. Let’s
analyze what each of the four regions of the diagram represent.
I: These are women aged 18-49 that smoke, but don’t drink.
II: These are women aged 18-49 that smoke AND drink.
III: These are women aged 18-49, that drink, but don’t smoke.
IV: These are women aged 18-49 that neither drink nor smoke.
In this particular example we assume there is overlap between
the smokers and drinkers in our survey. However, sometimes
sets can be disjoint.
Def: Two disjoint sets are those that do not overlap and hence have no common elements.
A U B We would use a diagram like that to the left if we discovered no one both smoked
and drank.
U
Sometimes sets are equal. For example if we found out that all the women who smoked also drank, and vice versa,
our two sets would be exactly equal. We represent this on a Venn Diagram as:
U A = B Last, sometimes sets are proper subsets of one another. Recall the
definition of a proper subset. We represent this case on a Venn diagram as:
U they
those
II.
A B In this particular diagram notice the B set is completely within the A set. For
our example that would mean all the women who said they drank also said
they smoked. However the region outside of B, and still within A would be
women who said they smoked, but did not drink.
Set Operators
a. The Complement of a set
Def: The complement of set A, symbolized by A' is the set all elements in U, but NOT in A.
Ex. Let U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 2, 4, 8}. Find A'
We want everything in the universal set, but NOT in A. This would be: {3, 5, 6, 7}, so A' = {3,5,6,7}
b. The Intersection of sets
Def: The intersection of sets A and B written as A ∩ B is the set of all elements common to both set A AND B.
This would be region II in the earlier Venn diagram. Using the word “and” means intersection in math.
Ex. Let A = {2, 6, 10}, B = φ . Find A ∩ B
Here is what we are trying to find: A ∩ B = {2,6,10} ∩ φ = {2,6,10} ∩ {} . Since the empty set has no elements
in it, it can have NOTHING in common with any other set. If it has nothing in common, the intersection must
ALSO be the empty set. Therefore, A ∩ B = φ
c. The Union of Sets
Def: The union of sets A and B, written as A ∪ B , is the set of elements that are members of set A OR of set B OR
of both sets. Basically A ∪ B is all elements contained somewhere in either set A or set B. This would be regions I,
II, and III in the earlier Venn diagram. Using the word “or” means union in math.
Ex. Let A = {1, 2, 4 ,6}, B = {3, 5, 6, 7, 8}. Find A ∪ B
When we union two sets, we basically combine together (but don’t repeat any elements). Think of union like
two people getting married (with previous families) and the families forming a “union” under one roof. So we’ll
write each element that is in A, then list all the elements in B, but just make sure to not repeat anything.
A ∪ B = {1,2,3,4,5,6,7,8}
Ex. Let A = {1, 2, 4 ,6}, B = φ . Find A ∪ B
**You try this**
The examples using the empty set ( φ ) point out two common identities.
1. A ∪ φ = A (i.e. when you union a set with the null set, the set remains unchanged)
2. A ∩ φ = φ (i.e. when you intersect a set with the null set, the result is the null set)
III.
Performing Multiple Set Operations
In these examples you should be able to perform set operations if you are given the Venn diagram and also if you are
given the sets in roster method. Both types will be on HW and the test.
Order of operations:
1. Perform operations in parentheses first
2. Complements on a set are usually performed first followed by unions and intersections
Ex. Let U = {1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5, 7}, B = {1, 2, 3}, C = {2, 3, 4, 5, 6}
a. Find B ∪ C '
b. Find ( B ∩ C )'
c. Find ( A ∪ B ) ∩ A'
a. I’ll do a.
1. First find C’. These are all the elements in U, but NOT in C. C ' = {1,7}
2. Now union B with what we got in step 1: B ∪ C '= {1,2,3,7}
You try b and c.
Answers: b. {1, 4, 5, 6, 7}
c. {2}
A 1 4
Ex.
3 7
B 2 5 6 8
9
U
Using the Venn diagram, find the following:
a. A'
b. A ∪ B '
c. ( A ∩ B )'
d. n( A ∪ B )
a. A’ would be any element outside of circle A: {2, 5, 6, 8, 9}
b.
A = {1, 3, 4, 7}, B’ = {1, 4, 8, 9},
c.
**You try this**
A ∪ B ' = {1,3,4,7,8,9}
Answer: {1, 2, 4, 5, 6, 8, 9}
d. Recall that the “n” operator wants the number of elements in that set (not the actual listed elements, just
how many there are)
Recall that ( A ∪ B ) is anything this in set A or B, (regions I, II, III on our earlier Venn diagram). If we count
up the number of elements in these regions we get 7. So n( A ∪ B ) = 7 .
Be careful on HW and test problems and note whether they are asking for a set like in parts a-c. of this last example (in
which case you list the elements in roster method) or if they want the cardinality (as in part d of this last example), in
which case you just count up the NUMBER of elements in the given set and give the number as your final answer, not
the actual set.
IV.
The Cardinal number of the Union of Two Finite Sets
A I II B III IV U
Is there a way to find n( A ∪ B ) by adding up the n(A) + n(B). The answer is yes and no. We can do this, but we need
to take into account a VERY important fact.
-First what regions represent n( A ∪ B ) ? **I, II, III**
-n(A) describes the number of elements in set A. Which two regions of the Venn diagram does this represent? *I, II*
-n(B) describes the number of elements in set B. Which two regions of the Venn diagram does this represent? *II, III*
Notice that if we add together n(A) + n(B) we get (I + II) + (II + III) = I + 2II + III. In effect we are COUNTING the
middle region (region II) twice!!! So since n( A ∪ B ) only needs 1 region I, 1 region II, and 1 region III, we have one
extra region II. To compensate, we subtract region II off of the sum.
Now, how do we describe region II using set notation? Region II is the intersection and is denoted by A ∩ B .
subtracted to compensate
So, the formula is then: n( A ∪ B ) = n( A) + n( B ) −
6474
8
n( A ∩ B )
Ex. Set A contains 30 elements, set B contains 18 elements, and 5 elements are common to sets A and B. How many
elements are in A ∪ B ?
**Just plug into the above formula**
Answer: 43
Application: Using the graph below, find the set of toys requested by more than 40% of the boys or more than 10% of
the girls.
Answer:
Set of toys requested by more than 40% of the boys: {Toy cars/trucks}
Set of toys requested by more than 10% of the girls: {Dollhouses, domestic accessories, dolls, spacial-temporal toys,
sports equipment}
So since the problem stated “OR” we find the union of these two sets:
{Toy cars/trucks, dollhouses, domestic accessories, dolls, spacial-temporal toys, sports equipment}