Sets in Mathematics
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DEFINITION 1
A
set
is an unordered collection of objects.
DEFINITION 2
The objects in a set are called the
elements
, or
members
, of the set. A set is said to contain
its elements.
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EXAMPLE 2
The set
O
of odd positive integers less than 1 0 can be expressed by
O
= {……………….}.Set builder notation:
•
the set 0 of all odd positive integers less than 1 0 can be written
as
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Some Important Sets
Some Important Sets
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N
= {O, 1 , 2, 3 , . . . } , the set ofnatural numbers
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Z
= { . . . , -2, - 1 , 0, 1 , 2, . . . } , the set of integers
•
z+
= { I , 2, 3, . . . } , the set of positive integers
•
Q
= {p/q | p ε
Z, q
ε
Z, and q ≠0 } , the set of
rational numbers
•
R
, the set of
real numbers.
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DEFINITION 4
THEOREM 1
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DEFINITION 5
Let S be a set. If there are exactly n distinct
elements in S where n is a nonnegative integer,
we say that S is a
finite set
and that n is the
cardinality of S
. The cardinality of S is denoted by
I S I
.
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EXAMPLE 9
Let A be the set of odd positive
integers less than 1 0.
EXAMPLE 10
Let S be the set of letters in the
English alphabet.
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DEFINITION 6
A set is said to be infinite if it is not finite.
EXAMPLE 12
The set of positive integers is infinite.
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The Power Set
The Power Set
DEFINITION 7
Given a set S, the
power set
of S is the set of all
subsets of the set S. The power set of S is
denoted by
P(S).
EXAMPLE 13
What is the power set of the set {0, 1 , 2} ?L
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If a set has n elements, then its power set has
2
n
n
elements.
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DEFINITION 8
DEFINITION 8
The ordered n-tuple (a
I
, a
2
, . . . , a
n
) is the
ordered collection that has a
l
as its first
element, a
2
as its second element, . . . , and a
n
as its n
th
element.
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آ
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ك
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We say that two ordered n -tuples are
equal
if and only if each
corresponding pair of their elements is equal.
•
In other words, (a
I
, a
2
, . . . , a
n
) = (b
l
, b
2
, • . • , b
n
) if and only if a
i
= b
i
, for i = 1 , 2, . . . , n .
•
In particular, 2-tuples are called ordered pairs. The ordered pairs
(a,b) and (c,d) are equal if and only if
a =c and b = d.
•
Note that (a,b) and (b,a) are not equal unless
a = b.
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The Cartesian product of two sets
The Cartesian product of two sets
DEFINITION 9
Let A and B be sets.
The Cartesian product
of A
and B, denoted by
A x B
, is the set of all
ordered pairs (a,b), where a
ϵ
A and b
ϵ
B .
Hence,
A x B = {(a , b) | a ϵ
A
ᴧ
b
ϵ
B } .
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EXAMPLE 16
What is the Cartesian product of A = { 1 , 2} andB= {a , b, c}?L
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Caution!
Caution!
The Cartesian products A x B and B x A
are not
equal
, unless
•
A = 0 or B= 0 (so that A x B = 0)
or
•
A = B
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The Cartesian product of sets
The Cartesian product of sets
DEFINITION 10
The
Cartesian product
of the sets A
1
, A
2
, . . . , A
n
,
denoted by A
1
x A
2
X • • • x A
n
, is the set of ordered n-
tuples (a
1
, a
2
, . . . , a
n
),
where a
j
belongs to A
j
for i =1 , 2 , . . . , n .
•
In other words
A
1
X A
2
x · · · x A
n
= {a1
, a
2
, . . . , a
n
) | a
j
ϵ
A
i
for i=1 ,2,.. ,n }.
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EXAMPLE 18
What is the Cartesian product A x B x C , where A =
{0, 1 }, B = { 1 , 2}, and C = {0,1 , 2}?L
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Page 119:
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5(a,b,c,d,f)
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7(a,b,d,f)
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Sets in mathematics are collections of objects where the order does not matter, and elements are unique. This concept explores the definition of sets, examples, important sets like natural numbers, integers, and rationals, equality of sets, cardinality, finite and infinite sets, and the power set. Sets play a fundamental role in various mathematical concepts and applications.
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Presentation Transcript
2.1- Sets 1 L Al-zaid Math1101
DEFINITION 1 A set is an unordered collection of objects. DEFINITION 2 The objects in a set are called the elements, or members, of the set. A set is said to contain its elements. 2 L Al-zaid Math1101
EXAMPLE 2 The set O of odd positive integers less than 1 0 can be expressed by O = { .}. Set builder notation: the set 0 of all odd positive integers less than 1 0 can be written as 3 L Al-zaid Math1101
Some Important Sets N = {O, 1 , 2, 3 , . . . } , the set of natural numbers Z = { . . . , -2, - 1 , 0, 1 , 2, . . . } , the set of integers z+ = { I , 2, 3, . . . } , the set of positive integers Q = {p/q | p Z, q Z, and q 0 } , the set of rational numbers R, the set of real numbers. 4 L Al-zaid Math1101
DEFINITION 3 Two sets are equal if and only if they have the same elements. That is, if A and B are sets, then A and B are equal if and only if ?(? ? ? ?) We write A = B if A and B are equal sets. EXAMPLE 6 The sets { 1, 3 , 5 } and { 3 , 5 , 1 } are equal, because they have the same elements. Remarks: Note that the order in which the elements of a set are listed does not matter. Note also that it does not matter if an element of a set is listed more than once, so { 1 , 3 , 3 , 3 , 5 , 5 , 5 , 5 } is the same as the set { 1 , 3 , 5 } because they have the same elements. 5 L Al-zaid Math1101
DEFINITION 4 THEOREM 1 6 L Al-zaid Math1101
DEFINITION 5 Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by I S I. 7 L Al-zaid Math1101
EXAMPLE 9 Let A be the set of odd positive integers less than 1 0. EXAMPLE 10 Let S be the set of letters in the English alphabet. 8 L Al-zaid Math1101
DEFINITION 6 A set is said to be infinite if it is not finite. EXAMPLE 12 The set of positive integers is infinite. 9 L Al-zaid Math1101
The Power Set DEFINITION 7 Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S). EXAMPLE 13 What is the power set of the set {0, 1 , 2} ? 10 L Al-zaid Math1101
Remark: If a set has n elements, then its power set has 2nelements. 11 L Al-zaid Math1101
Cartesian Products DEFINITION 8 The ordered n-tuple (aI, a2, . . . , an) is the ordered collection that has alas its first element, a2as its second element, . . . , and an as its nthelement. 12 .
Equality of two ordered n-tuples We say that two ordered n -tuples are equal if and only if each corresponding pair of their elements is equal. In other words, (aI, a2, . . . , an) = (bl, b2, . , bn) if and only if ai = bi, for i = 1 , 2, . . . , n . In particular, 2-tuples are called ordered pairs. The ordered pairs (a,b) and (c,d) are equal if and only if a =c and b = d. Note that (a,b) and (b,a) are not equal unless a = b. 13 L Al-zaid Math1101
The Cartesian product of two sets DEFINITION 9 Let A and B be sets. The Cartesian product of A and B, denoted by A x B, is the set of all ordered pairs (a,b), where a A and b B . Hence, A x B = {(a , b) | a A b B } . 14 L Al-zaid Math1101
EXAMPLE 16 What is the Cartesian product of A = { 1 , 2} and B= {a , b, c}? 15 L Al-zaid Math1101
Caution! The Cartesian products A x B and B x A are not equal, unless A = 0 or B= 0 (so that A x B = 0) or A = B 16 L Al-zaid Math1101
The Cartesian product of sets DEFINITION 10 The Cartesian product of the sets A1, A2, . . . , An, denoted by A1x A2X x An, is the set of ordered n- tuples (a1, a2, . . . , an), where ajbelongs to Ajfor i =1 , 2 , . . . , n . In other words A1X A2x x An= {a1, a2, . . . , an) | aj Aifor i=1 ,2,.. ,n }. 17 L Al-zaid Math1101
EXAMPLE 18 What is the Cartesian product A x B x C , where A = {0, 1 }, B = { 1 , 2}, and C = {0,1 , 2}? 18 L Al-zaid Math1101
Homework Page 119: 1 (a,b) 2(a) 4 5(a,b,c,d,f) 7(a,b,d,f) 9 19 L Al-zaid Math1101
