Set hw

The Basic Language of Sets
Exercise A. Farmer Andy’s family

Answer the following questions
using Farmer Andy’s family photo.
1. List the following sets and find Andy Beth Cathy Dan
the order of each.
R = {a member that is wearing something red}
G = {a member that is wearing something green}.
B = {a member that is wearing something blue}
2. List the following sets and find the order of each.
RH = {a member that has red hair}
BH = {a member that has black hair}
CG = {a child that is wearing something green}.
3. Using the above sets, answer the following as true or false.
a. c ϵ R b. d ϵ RH c. a G d. a ϵ G e. {a} G
∩

∩ ∩
e. a ϵ BH f. {b} ϵ RH g. {b} CG h. d ϵ B i. {d} BH
∩


R = {wearing something red}
G = {wearing something green}.
B = {wearing something blue}
Andy Beth Cathy Dan

List the following set, find the order of each and answer the
T or F questions.
4. W = {wearing both something red and something green}
W R? R W? W G? G W? Explain your answer.

∩
∩

∩
∩

5. V = {wearing something red or something green, or both}
R V? V R? G V? V G? Explain your answers.
∩

∩

∩

∩
6. X = {wearing something blue or something green, or both}
X B? B X? X G? G X? Explain your answer.
∩

∩

∩

∩

7. Y = {wearing something blue and something green}
B Y? Y B? Y G? G Y? Explain your answer.
∩

∩

∩
∩


R = {wearing something red}
G = {wearing something green}.
B = {wearing something red}
Andy Beth Cathy Dan

Recall that a sample of size n from a set A is a subset of n
elements selected from A.
8. G has 2 elements. List all the possible size 1 sample(s)
of G, then list all size 2 sample(s)
Why can’t there be any size 3 sample?
9. R has 3 elements. List all the possible size 1 sample(s)
of G, then list all the size 2 sample(s) and the size 3 sample(s).
Why can’t there be any size 4 sample?
10. If Farmer Andy’s family decide to take all the possible
photos consist of exactly three members of the family,
how may photos would be taken?
List the members in each photo.

Let the universal set U = {0, 1, 2,.., 9}.
List and find the order of each of the following subsets of each.
Express each answer as an inequality.
11. A = {x | x is more than 6}
12. B = {x | x is no less than 6}
13. C = {x | x is less than 3}
14. D = {x | x is no more than 3}
15. E = {x | x is even}
Following sample measurements are taken, classify each
measurement is discrete or continuous.
16. Your weight w 17. Your height h
18. The number of your siblings n
19. The amount of time you took to do last night’s homework t
20. The number of times you washed your hands today f

More on the Language of Sets
Exercise A. From the photo of Farmer Andy’s family in their new
dresses, we define the following sets
U = { the family} Farmer Andy’s family in new dresses

R = {red}
G = {green}
B = {blue}
F = {female}
M = {male} Andy Beth Cathy Dan
T = {has triangular decoration} C = {has circular decoration}
Exercise A.
1. a. List each set defined above and pick out all the mutually
exclusive pairs by inspection.
List the elements of each of the the following sets.
Name and describe each set in the form of {x | x is ###}.
b. FC c. MC d. CC e. TC f. GC g. UC
h. T U F i. F ∩ M j. B U C k. C U FC l. GC ∩ T k. TC ∩ CC

2. From Example C. we have the table as shown.
Let A = {Adult} ,T = {Teen}, Adult Teen Child
C = {Child}, F = {Female}, Male 5 17 18
M = {Male}, Female 15 26 19

Which pair of sets are mutually exclusively?
Name and describe each of the following sets in the form of
{x | x is ###} and find its order.
b. TC c. CC d. MC e. FC f. AC
g. A U F h. T ∩ C i. F ∩ MC j. AC U C k. CC U FC
3. Given intervals I, J , and K, do the set operation. Draw. Put the
answer in the interval and the inequality notations.
I: –8 –2 0 J: x < –3½ K: –5 ≤ x < 2
a. K U J b. K ∩ I c. I U J d. K U I
e. I ∩ J f. K ∩ J g. (K ∩ J) U I h. (K U J) ∩ I

4. Given following shapes and their areas, find the total area if we glue
them together as shown.

a.
10 cm2 20 cm2

A B AUB

b.
8 12
A B AUB
c. Each overlap is 1
A 20 A
B 20
C B
AUBUC
C 20

5. Farmer Andy produces nuts and sells them in three types of bags.
These are: peanuts only, cashews only, and
mixed peanuts–cashews. We bought many bags of each type.
Draw a diagram that reflects each of the following questions. Answer
the question if possible. If not, explain why not.
a. Out of the bags we bought, 30 contain peanut and 12 contain
cashews only. How many bags we bought in total?
How many peanuts only bags did we buy?
How many mixed begs did we buy?
b. Out of the bags we bought, 30 contain peanut only and 12
contain cashews only. How many bags we bought in total?
c. Out of the bags we bought, 30 contain peanut, 12 contain cashews
and there are 6 of mixed type. How many bags did we buy in total
and how may of each type did we buy?
d. Out of the 40 bags we bought, 30 contain peanut and 12 are of
mixed type. How many bags of each type did we buy?

6. Everyone in an extended family of 100 people have either genetic
traits A or B, or both. Answer each of the following questions with the
given information. Define and draw each set.
a. There are 85 people who have trait A. Out of the people with
trait A , 27 people have both traits A and B. How many people of
each type did we have?
b. There are 85 people who have trait A and there 27 people with B.
How many people of each type did we have?
c. Is it possible to have 85 people not having trait A and there 15
people with both traits A, B? What may we conclude?
d. Is it possible to have 85 people who don’t have trait B and
there 16 people with both traits A, B? What may we conclude?

Trees and Factorials
Exercise A.
There are multiple buses from the park to the mall where we may
change bus and go to downtown. Draw the tree diagram of all the
possible bus trips from the park to downtown given the following
information.
Red bus, Green bus, Bus No. 3
1 . park Blue bus, mall Bus No. 7
downtown

Red bus, Green bus, Bus No. 3
2 . park Blue bus, Yellow bus mall Bus No. 7 downtown
3. Continuing with problem 1, after downtown we are going
straight home either by the metro train or have Joe pick us up.
Draw the tree that represents all the possible trips from the park
to downtown, then home.
4. How many different bus trips are possible if we are going
from the park to downtown and back to the park. Do not draw
the tree.

5. Dan wants to take a math class in the morning then go directly to
a PE class. There are five choices for the math classes and for PE,
Dan can like do swimming, rock climbing or hole digging. How many
different possible morning class schedules Dan can have?
6. The Banana Inc. produces and sell tablets with different
configurations. There are 4 possible storage capacities:
100 TB, 10 TB, 1 TB, or 500 GB, three different types of processors:
the BS–super X, BS–super XX, and the Double BS. The tablets come in
black, white, and zebra stripes. How may different types of Banana
tablets are there?
7. This Friday night, Farmer Andy’s family can go out to eat
at a one out of the local five restaurants or eat yesterday’s
leftover, afterwards they can go to one of the three movies
or to the library; after that they can either go for boat ride or
paint the garage. How many different arrangements are
possible for Friday nights? Which is the least desirable?

9. For Cathy, an outfit consists of a hat, a shirt, a pair of pant
or a skirt, and a pair of shoes. If Cathy has 3 hats, 3 pairs of
shoes, 4 skirts, 4 shirts, and 5 pants, does Cathy have
enough to have a different outfit for each day of the year?
10. At the Marx’s Senior hamburger store, a Junior combo
special meal consists of a choice of a Lardy Burger, with or
without cheese, with or without bacons, a choice of an order of
fry Cheese balls or fry cookie, then a small diet C or diet P for
drink. How many different Junior combos are possible?
11. The music directory in Maria’s computer has 12
subdirectories for 12 types of music, in each subdirectory there
are 12 artists’ recordings and for each artist there are 12
songs stored. How many songs does Maria have?
12. Frank is writing a math book which consists of 8 chapters,
each chapter has 6 sections and each section has 30
homework problems, how many homework problems does
Frank have to come up with?

13. The main trunk of an Andy–berry tree, grows and splits into
3 boughs, each bough splits into 3 more branches, each branch
splits into 2 twigs, at the end of each twig it produce 5 small
berries, how many Andy–berries are there on one tree?
Suppose each acre can grow 120 trees, and Farmer Andy’s
has 80 acres of the berry–trees, and each berry can be sold
for 5¢, how much money will Andy earn if he harvest and sell
each one of them?
14. What’s 0!? 1!? 2!? 3!? 4!? 5!? 6!? 7!? 8!?
Calculate the following by cancelling as much as possible first.

15. Find a. 3! b. 4! c. 5! d.
7!
e.
8!
2! 2! 3! 4! 3!
3!
16. Find a. b. 4! c. 5! d. 7! e. 8!
2!1! 2!2! 2!3! 4!3! 3!5!
12!
17. Find a. 5!7! b. 14! c. 25! d. 17! e. 18!
12!2! 22!3! 9!5! 9!9!

Permutations and Combinations
Translate each of the following expressions into the nPr
notation or the factorial notation, then calculate the outcome
by cancelling as much as possible first.
1. a. 3! b. 3P2 c. 5! d. 6P2 e. 8!
2! 3! 3!
2. a. 3P1 b. 3! c. 5P2 d. 6! e. 8P5
1! 4!
Translate each of the following expressions into the nCr
notation or the factorial notation, then calculate the outcome

3. a. 3! b. 3C1 c. 5! d. 6 C2 e. 8!
2!1! 2!3! 3!5!
4. a. 3C2 b. 3! c. 5C2 d. 6! e. 8C5
1!2! 2!4!
5. Farmer Andy’s family {a, b, c, d} is Farmer Andy’s family
taking family photos. They are to sit in
four chairs in a row. How many different
sitting arrangements are possible? Andy Beth Cathy Dan

6. For the dinner cleaning duty, one member of Farmer Andy’s
family does the dishes and another member dries them.
How many different cleaning arrangements are there? List them.
7. From 6, if we meet a third person to take out the garbage,
how many different after dinner cleaning arrangements are
possible?
8. For the after dinner cleaning duty, two members of Farmer
Andy’s family are in charge of dish duty. How many different
after dinner cleaning 2–person teams are possible? List them.
9. From 8, if we need three members of Farmer Andy’s family in
charge of dish&garbage duty. How many different 3–person
teams are possible? List them.
10. Farmer Andy’s family are to draw lots to determine the
order of the bathroom usages in the morning. How many
different lineups of the bathroom usage are possible?
(Everyone uses the only bathroom in the morning.)

11. From problem 6, if the dish–washing job may be done by
anyone but the drying job must be done by a boy, how many
arrangements are possible? List them.
(Hint: Fill the drying job first)
12. From problem 11, besides that the drying job must be done
by a boy, the dish–washing job has to be done by a girl,
how many arrangements are possible? List them.
13. From problem 6, if the dish–washing job and the drying
job must be done by members from different genders,
14. From 6, Andy could participate in the drying but
he can’t wash dishes because he is allergic to dish–washing,
15. Back to problem 13, besides that Andy is allergic to dish–
washing that Beth is allergic to dish–drying but can participate
in washing, how many arrangements are possible? List them.

16. From the set of digits D = {0, 1, 2, .. 9} we are to line up seven
different digits. How many different numbers we may have?
17. How many 7–digit phone numbers are possible if the first
digit can’t be 0 and that no digit is repeated?
18. How many 7–digit phone numbers are possible if digit
may be repeated?
digit can’t be 0 and that digit may be repeated?

A B
We have two boxes A and B each contains different gifts as
shown. Answer the following questions.
20. We are to select two items to keep, how many different
two–item gifts are possible?

21. We are to select three items to keep, how many different
three–item gifts are possible?
22. We are to select two items to keep but they must be from
different boxes, how many such two–item gifts are possible?
the same boxes, how many such two–item gifts are possible?
24. Why is the sum of the answers from 22 and 23 is the answer of
problem 19?
25. We are to select three items to keep but they must be from
the same boxes, how many such three–item gifts are possible?
26. We are to select three items to keep but they can’t be all
from box A, how many such three–item gifts are possible?
(Use the complementary counting principle)
from box B, how many such three–item gifts are possible?

Translate each of the following expressions into the nPr notation
/ or the factorial notation, then calculate the outcome
1. a. 3! b. 3P2 c. 5! d. 6P2 e. 8!
2! 3! 3!
2. a. 3P1 b. 3! c. 5P2 d. 6! e. 8P5
1! 4!
Translate each of the following expressions into the nCr notation
/ or the factorial notation, then calculate the outcome of each

3. a. 3! b. 3C1 c. 5! d. 6 C2 e. 8!
2!1! 2!3! 3!5!
4. a. 3C2 b. 3! c. 5C2 d. 6! e. 8C5
1!2! 2!4!
5. Farmer Andy’s family {a, b, c, d} is Farmer Andy’s family
taking family photos. They are to sit in
four chairs in a row. How many different
sitting arrangements are possible? Andy Beth Cathy Dan

6. For the dinner cleaning duty, one member of Farmer Andy’s
family does the dishes and another member dries them.
How many different cleaning arrangements are there? List them.
7. From 6, if we meet a third person to take out the garbage,
how many different after dinner cleaning arrangements are
possible? List them.
8. For the after dinner cleaning duty, two members of Farmer
Andy’s family are in charge of dish duty. How many different
after dinner cleaning 2–person teams are possible? List them.
9. From 8, if we need three members of Farmer Andy’s family in
charge of dish&garbage duty. How many different 3–person
teams are possible? List them.
10. Farmer Andy’s family are to draw lots to determine the
order of the bathroom usages in the morning. How many
different lineups of the bathroom usage are possible?
(Everyone uses the only one bathroom in the morning.)

11. From problem 6, if the dish–washing job may be done by
anyone but the drying job must be done by a boy, how many
arrangements are possible? List them.
(Hint: Fill the drying job first)
12. From problem11, besides that the drying job must be done
by a boy, the dish–washing job has to be done by a girl,
13. From problem 6, if the dish–washing job and the drying
job must be done by members from different genders,
14. From 6, Andy could participate in the drying but
he can’t wash dishes because he is allergic to dish–washing,
How many arrangements are possible? List them.
15. Back to problem 13, besides that Andy is allergic to dish–
washing that Beth is allergic to dish–drying but can participate
in washing, how many arrangements are possible? List them.

16. From the set of digits D = {0, 1, 2, .. 9} we to line up 7
different digits. How many different numbers we may have?
digit can’t be 0 and that no digit is repeated?
18. How many 7–digit phone numbers are possible if digit
may be repeated?
digit can’t be 0 and that digit may be repeated?

A B
We have two boxes A and B each contains different gifts as
shown. Answer the following questions.
20. We are to select two items to keep, how many different
two–item gifts are possible?

21. We are to select three items to keep, how many different
three–item gifts are possible?
different boxes, how many such two–item gifts are possible?
the same boxes, how many such two–item gifts are possible?
24. We are to select three items to keep but they must be from
the same boxes, how many such three–item gifts are possible?
25. Why is the sum of the answers from 23 and 25 is the
answer of problem 19?
from box A, how many such three–item gifts are possible?
from box B, how many such three–item gifts are possible?

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