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Lecture 1.2
Contemporary Mathematics
Instruction: Introduction to Sets
A set is a collection of distinct objects. The objects of a set are elements. In the next few
assignments, we will deal with sets whose elements are numbers, but the elements of a set can be
any type of object, be it people, dog breeds, shapes, office furniture, baseball cards, comics, etc.
For introductory purposes we will consider the set of letters in the English alphabet. We
will call this set U, the universal set. The universal set in a discussion or other context is the
"universe" so to speak. If we designate the universal set as the letters in the English alphabet, we
restrict the universe to letters in the English alphabet.
Using roster or list notation,
U = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}.
Roster notation can be cumbersome, so there are other types of notation, one being descriptive
notation:
U = {letters of the English alphabet}
The symbol ∈ means "is an element of" so that " p ∈ U " reads "p is an element of U." The
symbol ∉ means "is not an element of" so that "π ∉ U " reads "pi is not an element of U."
Every set has a cardinal number. The cardinal number of a set, n, equals the number of
elements in the set. We denote the cardinal number of U as n (U ) . If U = {the letters of the
English alphabet}, then n (U ) = 26 .
A set without elements is called an empty or null set and denoted { } or ∅ . Note that ∅
is not equivalent to zero.
A subset is a set whose elements all appear in some other set. Set A is a subset of B if B
contains all the elements of A. The empty set is a subset of any set. If S = {the true vowels of
the English language}, then S is a subset of U since all vowels of the English language appear in
the set of letters of the English language. If T = {a, e, i}, then T is a subset of S since all the
elements in T appear in S. If V = {a, e, i, o, u} then V is a subset of S since every element of V
appears in S. Likewise, S is a subset of V since all the elements in S appear in V. Any set has 2n
subsets. Since n(S) = 5, then S has 25 = 32 subsets. Since n(T) = 3, then T has 23 or 8 subsets.
The subsets of T are: {a, e, i}, {a, e}, {a, i}, {e, i}, {a}, {e}, {i}, { }.
Two sets are equal if they each contain exactly the same elements. Or, alternatively, two
sets are equal if each of the sets is a subset of the other. Since V and S are subsets of one another
V = S. If T = {a, e, i} and R = {i, e, a}, then T = R.
Two equal sets are subsets of one another but not proper subsets. Set A is a proper subset
of B if A has fewer elements than B and B contains all the elements of A. The symbol ⊂ means
proper subset while ⊆ means subset so that the notation A ⊂ B reads "A is a proper subset of B"
while D ⊆ E reads "D is a subset of E." The symbol ⊄ means not a proper subset so that
F ⊄ G reads "F is not a proper subset of G." Alert readers may note the similarity between the
proper subset symbol ⊂ and a less than symbol < while noting that the subset symbol ⊆ shares
some similarity with the less than or equal to symbol ≤ .
Applications Exercise 1.2
Problems
Consider the two sets S and D given below.
S = {America's five space shuttle orbiters designed for space flight}
D = {Challenger, Apollo 1}
#1
Use roster notation to describe a non-empty set V if V ⊆ S and V ⊆ D .
#2
What is n ( S ) ?
#3
Referring to problem one, what is n(V)?
#4
How many subsets does S have?
#5
How many proper subsets does S have?
#6
Write an expression for the number of proper subsets of a set with n elements.
1. V = {Challenger}
2. n ( S ) = 5
3. n (V ) = 1
4. S has thirty-two subsets.
5. S has thirty-one proper subsets.
6. 2n − 1
Assignment 1.2
Problems
#1
Consider that B = {a, e, i, o} and C = {o, u}. Use roster notation to describe a non-empty
set D if D ⊆ B and D ⊆ C .
#2
Referring to problem one, what is n(D)?
#3
If X = {w, x, y, z}, then how many subsets does X have?
#4
Referring to problem three, how many proper subsets does X have?
#5
Write an expression for the number of proper subsets of a set with n elements.
Lecture 1.3
Contemporary Mathematics
Instruction: Set Operations
This lecture discusses set operations including complement, difference, intersection,
union, and Cartesian product. For this discussion, we will assume that U is the set of letters in
the English alphabet. Using this universal set, we will consider three subsets of U described
below in roster notation.
X = {a, e, i, o, u}, Y = {e, i, y}, and W = {u, v, s}.
The complement of a set S is the set of elements in U, not in S. We denote the
complement of set S with S C or S ' (or even S ). Thus,
X C = the complement of X = {the consonants of the English language}.
Similarly,
Y C = complement of Y = {a, b, c, d, f, g, h, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, z}
We use a minus sign to indicate the difference of two sets. The difference X – Y is the
set of elements that are elements in X but not elements in Y, e.g.,
X – Y = {a, o, u} and Y – X = {y}
Note that Y − X ≠ X − Y .
A second set operation is the union of sets. The symbol ∪ indicates union so that A ∪ B
reads "the union of set A with set B." The union of a two sets contains the elements of A as well
as the elements of B, e.g.,
X ∪ Y = {a, e, i, o, u, y}
The definition of union implies that A ∪ B = B ∪ A in general.
Inverting the union symbol ∪ creates the intersection symbol ∩ . The notation
A ∩ B reads "the intersection of set A and set B." The intersection of two sets contains the
elements common to both sets, e.g.,
X ∩ Y = {e, i}
The definition of intersection implies that A ∩ B = B ∩ A in general.
With set operations, multiple operations are possible as in the following example.
( X − W ) ∩ ( X ∪ W ) = ({a, e, i, o, u} − {u, v, s}) ∩ ({a, e, i, o, u} ∪ {u, v, s})
= {a, e, i, o} ∩ {a, e, i, o, u, v, s}
= {a, e, i, o}
Lecture 1.3
Multiple operations can also involve the complement of a set as below.
( X ∪ Y ∪W )
C
∪ (Y − W ) = ({a, e, i, o, u} ∪ {e, i, y} ∪ {u, v, s}) ∪ ({e, i, y} − {u, v, s})
C
= ({a, e, i, o, u, y, v, s}) ∪ {e, i, y}
C
= {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, w, x, z} ∪ {e, i, y}
= {b, c, d, e, f, g, h, i, j, k, l, m, n, p, q, r, s, t, w, x, y, z}
A Cartesian product is another set operation. The Cartesian product of two sets Y and W,
denoted Y × W , is the set of all possible ordered pairs whose first component is an element from
Y and whose second component is an element from W, e.g.,
Y × W = {(e,u), (e,v), (e,s), (i,u), (i,v), (i,s), (y,u), (y,v), (y,s)}
Note that the Cartesian product of two sets is a set whose elements are ordered pairs.
The definition of a Cartesian product can be extended, however, to include the product of
more than two sets. The Cartesian product of three sets Y, W, and Z is the set of all possible
ordered triples whose first component is an element from Y, whose second component is an
element from W, and whose third component is an element from Z. It follows that the Cartesian
product of three sets is a set of ordered triples. Likewise, the Cartesian product of four sets is a
set of ordered quadruples, and the Cartesian product of p sets is a set of ordered p-tuples.
To generalize the definition of a Cartesian product of two or more sets, consider a set, S,
whose elements are sets: { X 1 , X 2 ,… , X p −1 , X p } . The Cartesian product of the elements of S is
the set of ordered p-tuples defined below.
X1 × X 2 ×
× X p −1 × X p =
{( x , x ,…, x
1
1
p −1
, x p ) x1 ∈ X 1 , x2 ∈ X 2 , … , x p −1 ∈ X p −1 , x p ∈ X p
}
The Cartesian product is named after Rene Descartes, the seventeenth-century French
mathematician who analyzed mathematical objects using a coordinate system like the one below.
This coordinate system—formed by the intersection of a horizontal number line or axis with a
vertical number line or axis—is frequently called a Cartesian plane as well as a coordinate plane
Lecture 1.3
and/or x-y plane. An ordered pair of real numbers—that is a pair of real numbers in x-y specific
order—represents a point on the Cartesian plane. The ordered pair (2,2) is shown below.
Application Exercise 1.3
Problems
Consider the universal set: U = {Apollo mission designations 1 through 17} , and these subsets:
D = {disastrous Apollo missions causing fatalities to crew }
E = {unused Apollo mission designations}
M = {manned Apollo missions}
N = {unmanned Apollo missions}
T = {Apollo missions six through nine}
S = {Apollo missions eight through twelve}
Use roster notation to describe the following sets.
#1
Find T C .
#2
Find T ∪ S .
#3
Find T ∩ S .
#4
(E ∪ M )
#5
T −M
#6
(M ∪ N )
#7
E∩M
#8
D× E
C
1. T C = {1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 16, 17}
2. T ∪ S = {6, 7, 8, 9, 10, 11, 12}
4.
5.
7.
( E ∪ M ) = {1, 2, 3, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}
T − M = {6}
E ∩M ={ }
3. T ∩ S = {8, 9}
6. ( M ∪ N ) = {2, 3, 4}
C
8. D × E = {(1, 2 ) , (1,3)}
Assignment 1.3
Problems
Consider the universal set: U = { , , , ○, a, c, e, h, i, m, s, t, 1, 2, 3, 4, 5, 6} , and these subsets:
G = { , , , e} , K = { , h, m, s, t} , L = {a, c, e, h, i, m, s, t} , M = { , h, 2} , N = {4, 5, 6} ,
S = {1, 2, 3} , & V = {a, e, i} .
#1
Find LC .
#2
Find M ∪ S .
#3
Find M ∩ S .
#4
(G ∪ K ) ∩ ( L − V )
#5
Find S × N .
Challenge Point Problem
Consider two sets, X and Y, such that X = {all the real numbers} and Y = {all the real numbers}.
What mathematical object can be used to represent X × Y ?