Contemporary Mathematics

Chapter 1

1.1
1.
One possible solution: $T = \{ {\text{wrench, screwdriver, hammer, plyers}}\}$.
1.2
1.
This is not a well-defined set.
2.
This is a well-defined set.
1.3
1.
$\emptyset {\text{ or \{ \} }}$
1.4
1.
$D = \{ 0,1,2, \ldots ,9\}$
1.5
1.
$M = \{ 1,3,5, \ldots \}$
1.6
1.
$C = \{c|c{\text{ }}{\text{is a car}}\}$
1.7
1.
$I = \{i|i{\text{ }}{\text{is a musical instrument}}\}$
1.8
1.
$n(P) = 0$
2.
$n(A) = 26$
1.9
1.
finite
2.
infinite
1.10
1.
Set $B$ is equal to set $A$, $B = A$
2.
neither
3.
Set $B$ is equivalent to set $C$, $B \sim C$
1.11
1.
$\{ {\text{heads, tails}}\} ;$$\{ {\text{heads}}\} ,{\text{ }}\{ {\text{tails}}\} ;$ and $\emptyset$
1.12
1.

A set with one member could contain any one of the following:

$\{ {\text{Articuno}}\} {\text{, }}\{ {\text{Zapdos}}\} {\text{, }}\{ {\text{Moltres}}\} {\text{, or }}\{ {\text{Mewtwo}}\}$.

2.

Any of the following combinations of three members would work:

$\{ {\text{Articuno, Zapdos, Mewtwo}}\}$,$\{ {\text{Articuno, Moltres, Mewtwo}}\}$, or $\{ {\text{Zapdos, Moltres, Mewtwo}}\}$.

3.
The empty set is represented as $\{ {\text{ }}\}$ or $\emptyset$.
1.13
1.
$E \subset \mathbb{N}$
1.14
1.
512
1.15
1.
$\{ m|m = 5n{\text{ where }}n \in \mathbb{N}\}$
1.16
1.

Serena also ordered a fish sandwich and chicken nuggets, because for the two sets to be equal they must contain the exact same items: {fish sandwich, chicken nuggets} = {fish sandwich, chicken nuggets}.

1.17
1.

There are multiple possible solutions. Each set must contain two players, but both players cannot be the same, otherwise the two sets would be equal, not equivalent. For example, {Maria, Shantelle} and {Angie, Maria}.

1.18
1.
The set of lions is a subset of the universal set of cats. In other words, the Venn diagram depicts the relationship that all lions are cats. This is expressed symbolically as ${L} \subset {U}$.
1.19
1.
The set of eagles and the set of canaries are two disjoint subsets of the universal set of all birds. No eagle is a canary, and no canary is an eagle.
1.20
1.
The universal set is the set of integers. Draw a rectangle and label it with $U = \text{Integers}$. Next, draw a circle in the rectangle and label with Natural numbers.
Venn diagram with universal set, $U=\text{Integers}$, and subset $\mathbb{N} = {\text{Natural numbers}}$.
2.
Venn Diagram with universal set, $U$ and subset $A$.
1.21
1.
Venn Diagram with universal set, $U =$ Things that can fly with disjoint subsets Airplanes and Birds.
1.22
1.
$A' = \left\{ {{\text{orange, green, indigo, violet}}} \right\}$
2.
$A' = \{ c \in U|c{\text{ is a lion}}\}$ or $A' = \{ c \in U|c \notin A\}$
1.23
1.
$A\mathop \cap \nolimits B = \{ a\}$
1.24
1.
$A \cap B = \{\,\,\}$
1.25
1.
$A \cap B = B = \{a,e,i,o,u\}$
1.26
1.
$A \cup B = \{a,d,h,p,s,y\}$
1.27
1.
$A \cup B = \{{\text{red, yellow, blue, orange, green, purple\}}}$
1.28
1.
$A \cup B = \{a, b, c,\ldots,z\} = A$.
1.29
1.

33

1.30
1.

113

1.31
1.
$A$ or $B = A \cup B = \{h, a, p, y, w, e, s, o, m\} .$
2.
$A$ and $C = A \cap C = \{a, h\} .$
3.
$B$ or $C = B \cup C = \{a, w, e, s, o, m, t, h\} .$
4.
($A$ and $C$) and $B = (A \cap C) \cap B = \{a, h\} \cap \{a, w, e, s, o, m\} = \{a\} .$
1.32
1.

127

2.

50

1.33
1.

$A \cap B = \{ 3,5,7\}$.

2.

$A \cup B = \{ 1,2,3,5,7,9\}$.

3.

$A \cap B' = \{ 1,9\}$.

4.

$n(A \cap B') = 2$.

1.34
1.
40
2.
0
3.
27
1.35
1.

$n(B) = n(A{B^ + }) + n(A{B^ - }) + n({B^ + }) + n({B^ - }) = 14$

2.

$n({B^\prime }) = n(U) - n(B) = 86$

3.

$n(B \cup R{h^ + }) = 87$

1.36
1.
Venn diagram – Attendees at a conference with sets: Soup, Sandwich, Salad – Complete Solution
1.37
1.
$A \cap (B \cap C) = \{ 0,1,2,3,4,5,6\} \cap \{ 0,6,12\} = \{ 0,6\}$
2.
$(A \cap B) \cup (B \cap C) = \{ 0,2,4,6\} \cup \{ 0,3,6\} = \{ 0,2,3,4,6\}$
3.
$(A \cup {C^\prime }) \cap (B \cup {C^\prime }) = \{ 0,1,2,3,4,5,6,7,8,10,11\} \cap \{ 0,1,2,4,5,6,7,8,10,11,12\} = \{ 0,1,2,4,5,6,7,8,10,11\}$
1.38
1.

The left side of the equation is:

Venn diagram of intersection of two sets and its complement.

The right side of the equation is given by:

Venn diagram of union of the complement of two sets.

1.
set
2.
cardinality
3.
not a well-defined set
4.
12
5.
equivalent, but not equal
6.
finite
7.

Roster method: $\{ \text{A, B, C, } \ldots \text{, Z}\},$ and set builder notation: $\{ x|x{\text{ is a capital letter of the English alphabet}}\}$

8.
subset
9.

To be a subset of a set, every member of the subset must also be a member of the set. To be a proper subset, there must be at least one member of the set that is not also in the subset.

10.
empty
11.
true
12.
${2^{10}} = 1024$
13.
equivalent
14.
equal
15.
relationship
16.
universal
17.
disjoint or non-overlapping
18.
complement
19.
disjoint
20.

intersection

21.

union

22.

$A \cup B$

23.

$A \cap B$

24.
$A$
25.
$B$
26.
empty
27.
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
28.

overlap

29.

central

30.
intersection of all three sets, $A \cap B \cap C$
31.

parentheses, complement

32.
equation, true
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