Game Mathematics Exercise Set.pdf

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Game Math Exercises, by Frank Luna
Game Math Exercises
Frank Luna
www.gameinstitute.com
The following exercises will help prepare you for the exams. In general, exam questions
will be like the following exercise questions, but with the numbers changed. Therefore,
if you can do the exercises well, then that is a good indicator that you will do well on the
exams. Answers to most problems are provided, but not solutions. If you need a
solution, or perhaps an example, then consult with your instructor. Note that you should
be doing the exercises from this exercise set and not the textbook’s exercises, except
when this exercise set specifically asks you to do a problem from the main textbook.
You will not be tested on material outside this exercise set.
Game Math Chapter 1 Exercises
Chapter Remarks: The main skill from this chapter that I want you to walk away with is
being comfortable with the terminology of sets, and how the basic operations work;
always have the Venn diagrams in mind to organize your thoughts when reasoning about
sets. At this point, also be familiar with the basic ideas of functions, which map elements
from one set to another; remember that to be a function, an element in the domain cannot
map to two different elements in the range, although two different elements in the domain
can map to the same range element. Also know the condition for invertibility, and
understand why this condition is necessary. Most of the topics covered in this chapter
are mainly groundwork for later chapters. The following exercises should test what I
expect you to be able to do after studying this chapter.
1. True or false. If false, give a counterexample. Assume A and B are sets.
bacc f d d de abcde f
,,,, ,,,,
} {
=
,,,,,
}
.
b. If {
Aabcdef
,,, ,,
}
, then cA
∉ .
∪=+.
d. If {
Aabcdef
,,, ,,
}
and { }
B abc
=
,,
, then AB
⊆ .
e. ]\ .
f. If AB
⊆ or B A
⊆ , then AB
= .
]] ] .
i. { }
=∪∪
+ −
{ 0
]] .
j. AB B A
=∩∩
+ −
{ }
−=−
- 1 -
a. {
=
c. AB A B
=
g. ∅⊆ \\ .
h.
0
0
736239401.003.png
Game Math Exercises, by Frank Luna
k. Suppose the universal set of discourse is ] . If { 0
A = , then
A
= ]] .
+ −
2. Definition: An integer n is odd if and only if we can write
= + , for some
21
integer k . Define the set of all odd integers.
3. Definition: An integer n is even if and only if we can write
nk
= , for some
integer k . Define the set of all even integers.
4. Prove the following simple facts about odd and even integers:
a. The sum of two odd integers is even.
b. The sum of two even integers is even.
c. The sum of an even integer and an odd integer is odd.
d. The product of two odd integers is odd.
e. The product of two even integers is even.
f. The product of even integer and an odd integer is even.
Example: We prove (a): Let
a =+ and
21
bn
= + , where mn ∈ ] . Then
21
+= ++ + Definition of a and b odd integers
222
( ) ( )
2121
n
=++
mn
Algebra
( )
=++
2
mn
1
Morealgebra
2 k
=
Set
kmn
= ++, for k ∈ ]
1
The number k is obviously an integer, so we have shown that the sum of two odd
integers has the form of an even integer
ab k
+ = (i.e., it is an even integer). Thus we
2
are done.
Remarks: Note that in this proof we kept things very general. We did not assume
specific values for a and b ; if we did, then our proof would not be valid for all
integers—it would only be valid for those specific integers chosen. In the proof, we pick
arbitrary odd integers a and b , and then just use the fact that they are odd integers to
move to the next step. The rest is just algebra until we show the resulting form is that of
an even integer. The key is, because we have shown it is true for arbitrary odd integers a
and b , it is true for all odd integers.
5. Let {
Uabcdef ghi j
=
,,, ,, , ,,,
}
be the universal set of discourse, and let
Aacf h
=
{ }
,, ,
and { }
B ch j
=
,,
. Find the following:
∪
b. AB
∩
c. AB
−
e.
U
C
- 2 -
nk
ab m
a. AB
−
d. B A
Game Math Exercises, by Frank Luna
C B
j. ( ) ( )
∪−∩
k. ( ) ( ) ( )
AB AB
AB B A A B
−∩−∩∩
6. Let {
U =
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
}
be the universal set of discourse, and let
A =
{ }
1, 2, 9, 1 0
, { }
B =
1, 2, 6, 8
, and { }
C =
1, 3, 5, 7 , 9
. Find the following:
∪∪
b. ABC
∩∩
c. U −
d. C
e. ( )
∪−
f. ( )
AB C
∩∪
g. ( ) ( )
ABC
∩∪∩
h. ( )
AB AC
∪∩
i. ( ) ( )
ABC
∪∩∪
j. ( ) C
AB AC
A ∩
k.
A ∪
C
C
7. Let C be the set which includes all the elements in A or B , but not both.
Express C in terms of the sets A and B . (Hint: A Venn diagram helps organize
your thoughts.)
8. Let A and B be sets. Express A ∪ as the union of three disjoint sets. (Hint: A
Venn diagram helps organize your thoughts.)
9. Do all the exercises in the textbook for this chapter. For #5, just draw a Venn
diagram to illustrate the fact.
Answers to Selected Problems
1a. True.
1b. False, c ∈ .
1c. False, suppose { }
Aabc
=
,,
and { }
B c = . Then
,
A ∪=, but
4
A = and
B = . Hence, AB A B
∪≠ +.
- 3 -
f. A
g. A
h. B
i. ( ) C
a. ABC
3
2
736239401.004.png 736239401.005.png 736239401.006.png 736239401.001.png
Game Math Exercises, by Frank Luna
1d. False, eA
∈ but eB
∉ .
1e. True.
1f. False, AB
= if and only if AB
⊆ and B A
⊆ .
]]
1j. False, suppose { }
+ ∩∩ =∅
{ 0
Aabc
=
,,
and { }
B cd
= . Then { }
,
−= but { }
B Ad
1k. True.
= = + ]
3. {
Sxxk k
:
2,
}
Sxxkk
= = ]
:
2,
}
5a. { }
AB acfhj
∪=
,, ,,
∩=
5c. { }
AB ch
,
−=
5d { }
AB af
,
B Aj
−=
5e.
U =∅
C
A =
5g. {
4
C Abde gi j
=
,,,,,
}
5h. {
B
C
=
abd e f gi
,, ,, , ,
}
=
5j. ( ) ( ) { }
B
C
B
∪−∩=
5k. ( ) ( ) ( )
AB AB afj
,,
AB B A A B
−∩−∩∩=∅
6a.
ABC
∪∪=
{
1, 2, 3, 5, 6, 7, 8, 9,10
}
∩∩=
6c. {
ABC
{ 1
U −=
3, 4, 5, 6, 7, 8
}
6d. { }
C =
C
2, 4, 6, 8,10
∪−=
6f. ( ) { }
AB C
∩∪=
6g. ( ) ( ) { }
ABC
1, 2, 9
∩∪∩=
6h. ( ) { }
AB AC
1, 2, 9
∪∩=
6i. ( ) ( ) { }
ABC
1, 2, 9,10
∪∩∪=
6j. ( ) {
AB AC
1, 2, 9,10
C
A ∩=
3, 4, 5, 6, 7, 8, 9,10
}
- 4 -
1g. True.
1h. True.
1i. False,
AB ab
,
−= .
2. {
5b. { }
5f.
5i. ( ) C
6b.
6e. ( ) 4
736239401.002.png
 
Game Math Exercises, by Frank Luna
6k.
A ∪=
C
{
3, 4, 5, 6, 7, 8, 9,10
}
=∪−∩. Draw a Venn diagram to see this!
8. ( ) ( ) ( )
CAB AB
AB AB BA AB
∪= − ∪ − ∪ ∩ . Draw a Venn diagram to see this!
- 5 -
C
7. ( ) ( )

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