2. What is a set?
◍ A set is a group of “objects”
◌ Classes offered by a department: { GE 1, GE
2, GE 3, … }
◌ Colors of a rainbow: { red, orange, yellow,
green, blue, indigo, violet }
◌ States of matter { solid, liquid, gas,
plasma }
◍ Although a set can contain (almost) anything,
we will most often use sets of numbers
◌ All positive numbers less than or equal to
5: {1, 2, 3, 4, 5}
◌ All real numbers: { 2.1, π, 0, -6.32, e,… }
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3. Set Properties 1
◍ Order does not matter
◌ {1, 2, 3, 4, 5} is
equivalent to {3, 5,
2, 4, 1}
◍ Sets are notated with
curly brackets
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4. Set properties 2
◍ Sets do not have duplicate elements
◌ Consider the set of vowels in the
alphabet.
It makes no sense to list them as
{a, a, a, e, i, o, o, o, o, o, u}
What we really want is just {a, e,
i, o, u}
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5. Specifying a Set 1
◍ Sets are usually represented by
a capital letter (A, B, S, etc.)
◍ Elements are usually represented
by an italic lower-case letter
(a, x, y, etc.)
◍ Easiest way to specify a set is
to list all the elements: A =
{1, 2, 3, 4, 5}
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6. Specifying a Set 2
◍ The roster method of specifying a set consists of
surrounding the collection of elements with braces.
A variation of the simple roster method uses the
ellipsis ( … ) when the pattern is obvious and the
set is large.
{1, 3, 5, 7, … , 9007} is the set of odd
counting numbers less than or equal to
9007.
{1, 2, 3, … } is the set of all counting
numbers.
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7. Specifying a Set 2
Set builder notation has the general form
{variable | descriptive statement }.
The vertical bar (in set builder notation) is always
The vertical bar (in set builder notation) is always
read as “such that”.
read as “such that”.
Set builder notation is frequently used when the
Set builder notation is frequently used when the
roster method is either inappropriate or
roster method is either inappropriate or
inadequate.
inadequate.
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8. EXAMPLE SET-BUILDER
NOTATION
{x | x < 6 and x is a counting number} is the set of
all counting numbers less than 6. Note this is the
same set as {1,2,3,4,5}.
{x | x is a fraction whose numerator is 1 and
whose denominator is a counting number }.
Set builder notation will become much more
Set builder notation will become much more
concise and precise as more information is
concise and precise as more information is
introduced.
introduced.
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9. Specifying a Set 3
◍ A set is said to “contain” the various
“members” or “elements” that make up the
set
◌ If an element a is a member of (or an
element of) a set S, we use then notation
a S
◌ 4 {1, 2, 3, 4}
◌ If an element is not a member of (or an
element of) a set S, we use the notation
a S
◌ 7 {1, 2, 3, 4}
◌ Virginia {1, 2, 3, 4}
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10. Often Used Sets
◍ N = {0, 1, 2, 3, …} is the set of natural
numbers
◍ Z = {…, -2, -1, 0, 1, 2, …} is the set
of integers
◍ Z+
= {1, 2, 3, …} is the set of positive
integers (a.k.a whole numbers)
◍ Q = {p/q | p Z, q Z, q ≠ 0} is the
set of rational numbers
◌ Any number that can be expressed as a
fraction of two integers (where the bottom
one is not zero)
◍ R is the set of real numbers
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11. The Universal Set
◍ U is the universal set – the set of all
of elements (or the “universe”) from
which given any set is drawn
◌ For the set {-2, 0.4, 2}, U would be
the real numbers
◌ For the set {0, 1, 2}, U could be the
natural numbers (zero and up), the
integers, the rational numbers, or
the real numbers, depending on the
context
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12. The Universal Set
◌ For the set of the students in this class,
U would be all the students in the
University
◌ For the set of the vowels of the alphabet,
U would be all the letters of the alphabet
◌ To differentiate U from U (which is a set
operation), the universal set is written
in a different font (and in bold and
italics)
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13. Venn Diagrams
◍ Represents sets graphically
◌ The box represents the universal set
◌ Circles represent the set(s)
◍ Consider set S, which is
the set of all vowels in the
alphabet
◍ The individual elements
are usually not written
in a Venn diagram
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a e i
o u
b c d f
g h j
k l m
n p q
r s t
v w x
y z
U
S
14. Sets of Sets
◍ Sets can contain other sets
◌ S = { {1}, {2}, {3} }
◌ T = { {1}, {{2}}, {{{3}}} }
◌ V = { {{1}, {{2}}}, {{{3}}},
{ {1}, {{2}}, {{{3}}} } }
◌ V has only 3 elements!
◍ Note that 1 ≠ {1} ≠ {{1}} ≠
{{{1}}}
◌ They are all different
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15. The Empty Set
◍ If a set has zero elements, it is called
the empty (or null) set
◌ Written using the symbol
◌ Thus, = { } VERY
IMPORTANT
◌ If you get confused about the empty set
in a problem, try replacing by { }
◍ As the empty set is a set, it can be a
element of other sets
◌ { , 1, 2, 3, x } is a valid set
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16. The Empty Set
◍ Note that ≠ { }
◌ The first is a set of zero elements
◌ The second is a set of 1 element
(that one element being the empty
set)
◍ Replace by { }, and you get: { } ≠
{ { } }
◌ It’s easier to see that they are
not equal that way
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17. Set Equality
◍ Two sets are equal if they have the
same elements
◌ {1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}
◌ Remember that order does not
matter!
◌ {1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2,
1}
◌ Remember that duplicate elements
do not matter!
◍ Two sets are not equal if they do not
have the same elements
◌ {1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}
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18. Subsets 1
◍ If all the elements of a set S are also
elements of a set T, then S is a subset of T
◌ For example, if S = {2, 4, 6} and T = {1, 2,
3, 4, 5, 6, 7}, then S is a subset of T
◌ This is specified by S T
◌ Or by {2, 4, 6} {1, 2, 3, 4, 5, 6, 7}
◍ If S is not a subset of T, it is written as
such:
S T
◌ For example, {0, 8, 9} {1, 2, 3, 4, 5, 6,
7}
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19. Subsets 2
◍ Note that any set is a subset
of itself!
◌ Given set S = {2, 4, 6},
since all the elements of S
are elements of S, S is a
subset of itself
◌ This is kind of like saying 5
is less than or equal to 5
◌ Thus, for any set S, S S
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20. Proper Subsets
◍ If S is a subset of T, and S is not equal to
T, then S is a proper subset of T
◌ Let T = {0, 1, 2, 3, 4, 5}
◌ If S = {1, 2, 3}, S is not equal to T, and
S is a subset of T
◌ A proper subset is written as S T
◌ Let R = {0, 1, 2, 3, 4, 5}. R is equal to
T, and thus is a subset (but not a proper
subset) or T
◌ Can be written as: R T and R T (or
just R = T)
◌ Let Q = {4, 5, 6}. Q is neither a subset
of T nor a proper subset of T
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21. Proper Subsets 2
◍ The difference between “subset”
and “proper subset” is like the
difference between “less than or
equal to” and “less than” for
numbers
◍ The empty set is a proper subset
of all sets other than the empty
set (as it is equal to the empty
set)
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22. Set Cardinality
◍ The cardinality of a set is the
number of elements in a set
◌ Written as |A|
◍ Examples
◌ Let R = {1, 2, 3, 4, 5}. Then |R| = 5
◌ || = 0
◌ Let S = {, {a}, {b}, {a, b}}. Then |S|
= 4
◍ This is the same notation used for
vector length in geometry
◍ A set with one element is sometimes
called a singleton set
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25. Set operations: Union
◍ Formal definition for the union of
two sets:
A U B = { x | x A or x B }
◍ Further examples
◌ {1, 2, 3} U {3, 4, 5} = {1, 2, 3,
4, 5}
◌ {New York, Washington} U {3, 4} =
{New York, Washington, 3, 4}
◌ {1, 2} U = {1, 2}
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27. Set operations: Intersection
◍ Formal definition for the intersection
of two sets: A ∩ B = { x | x A and x
B }
◍ Further examples
◌ {1, 2, 3} ∩ {3, 4, 5} = {3}
◌ {New York, Washington} ∩ {3, 4} =
◌ No elements in common
◌ {1, 2} ∩ =
◌ Any set intersection with the
empty set yields the empty set
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28. Content, images, text, etc. used belong to
the rightful owner. No copyright
infringement intended.
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Editor's Notes
#2:SET is a well defined collection of objects; the objects are called members or ELEMENTS
#7:A prime number is a whole number greater than 1 whose only factors are 1 and itself.
