Download SAT Tip #17 - Sets (with Answer Key)

Name _______________________________________________________________ Date ___________________
SAT Tip of the Week: SETS
A set is a group of things.
The members of a set are called its elements.
The union (u) of two or more sets is the set consisting of all of the elements of the united
sets.
The intersection (∩) of two or more sets is the set of the common elements.
Let set A = {1, 3, 5, 8, 9} and let set B = {2, 3, 4, 7, 8}
1. Write AuB: ____________________________
2. Write A∩B: _________________________
Let set K = {prime numbers less than 20}
3. Write A∩K: ________________________
4. Let X = the set of prime numbers and let Y = the set of multiples of three. Which set listed
below is the set of AuB?
a.
b.
c.
d.
e.
{0}
{3}
{0, 3}
{3, 9, 15}
The empty set
Venn Diagrams
A Venn diagram is a useful way to organize set information.
Let set M = the prime factors of 770
Let set N = the prime factors of 1092
5. List the elements of each set. M = {_______________________}, N = {_____________________}
6. List the elements of M∩N. _____________________
7. Display the sets using a Venn Diagram
4.
5.
6.
7.
1.
2.
3.
Answers:
AuB = {1, 2, 3, 4, 5, 7, 8, 9}
A∩B = {3, 8}
First K = {2, 3, 5, 7, 11, 13, 17, 19}
A∩K = {3, 5}
b. {3} – remember, 0 is not prime, and 9 is not prime
M = {2, 5, 7, 11}, N = {2, 3, 7, 13}
M∩N = {2, 7}
Name _______________________________________________________________ Date ___________________
Now, let’s put this together in a typical SAT style word problem.
Example #1: In a middle school class there are a
total of 40 students out of which 14 are in the
chorus, and 29 students are in band. There are 5
students that are in both chorus and band. How
many students are in neither chorus nor band?
Drawing a Venn diagram can be helpful:
Solution: To solve this problem we first need to find the number of students who are in only the
band and only the chorus. Simply subtract 29-5 = 24 and 14-5 = 9. Add this to the Venn
diagram.
So the number of students in band, chorus, or
both is 24+5+9 =38. That means, with a total of
40 students, there are 2 that are in neither band
nor chorus.
Example #2: In a particular 4th grade class, 11 children play spring baseball, and 13 play spring
soccer. If 16 of them play only one of the 2 sports, how many of them play both spring sports?
A) 1
B) 3
C) 4
D) 8
E) 10
So the number that play both is x. Just baseball is 11-x. Just soccer is 13-x. So just one
of the two sports is 11-x + 13-x = 16. x = 4. That means 4 play both. Answer: C) 4
Name _______________________________________________________________ Date ___________________
SAT Tip of the Week: SETS
Due _________
Directions: For each problem, you must show your work or explain your answer for full credit.
Remember, calculators are allowed on the entire SAT Math test. You must use a pencil.
1. The set S consists of all multiples of 6. Which
of the following sets are contained within
S?
I. The set of all multiples of 3
II. The set of all multiples of 9.
III. The set of all multiples of 12.
A.
B.
C.
D.
E.
I only
II only
III only
I and III only
II and III only
3. In a community of 416 people, each
person owns a dog or a cat or both. If
there are 316 dog owners and 280 cat
owners, how many of the dog owners are
not cat owners?
A.
B.
C.
D.
E.
36
100
136
180
316
2. At Best High School, the math club has 15
members and the chess club has 12
members. If a total of 13 students belong
to only one of the two clubs, how many
students belong to both clubs?
A. 2
B. 6
C. 7
D. 12
E. 14
4. For sets F and G, if the union of sets F and
G is the set of all positive factors of 36, and
set F = {2, 3, 6, 9, 18}, then which of the
following could be set G?
A.
B.
C.
D.
E.
{36}
{2, 4, 12, 36}
{4, 12}
{4, 12, 36}
{1, 4, 12, 36}
Name _______________________________________________________________ Date ___________________
SAT Tip of the Week: SETS
ANSWER KEY
Directions: For each problem, you must show your work or explain your answer for full credit.
Remember, calculators are allowed on the entire SAT Math test. You must use a pencil.
1. The set S consists of all multiples of 6.
Which of the following sets are contained
within S? C. III only
I The set of all multiples of 3
II The set of all multiples of 9.
III The set of all multiples of 12.
S = {6, 12, 18, 24, 36, …}
I contains a 3 which is not in set S
II contains 9 and 27 (and many more) which
are not in set S
3. In a community of 416 people, each
person owns a dog or a cat or both. If
there are 316 dog owners and 280 cat
owners, how many of the dog owners are
not cat owners? C. 136
Let x be the number that own both a cat and
a dog. 416 = 316-x + 280-x + x (the total 416 is
equal to the number of just dog owners, plus
the number of just cat owners, plus the number
that have both). We find that x = 180. So the
number of just dog is 316-180 = 136.
2. At Best High School, the math club has 15
members and the chess club has 12
members. If a total of 13 students belong
to only one of the two clubs, how many
students belong to both clubs? C. 7
Let x be the number in both clubs. For only
one club, 15-x + 12-x = 13; x=7
4. For sets F and G, if the union of sets F and
G is the set of all positive factors of 36, and
set F = {2, 3, 6, 9, 18}, then which of the
following could be set G? E. {1, 4, 12, 36}
F U G = {1, 2, 3, 4, 6, 9, 12, 36}
F = {2, 3, 6, 9, 18}
G must be the ones not in F so {1, 4, 12, 36}