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| Sets |
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Sets |
Definition
of a set and notation |
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Set
membership |
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Set
builder notation |
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Cardinal
number |
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Ordinal
number |
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Equal
sets |
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Subsets |
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The
empty set or the null set |
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Universal
set or universe |
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Power
set |
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| Definition
of a Set and Notation |
| Set
is a finite or infinite collection of distinct objects. Those
objects are called elements or members of the set. |
| The
notation, A
= {
a, b, c
}
means A equals
the set of elements a,
b
and c. |
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| Set
membership |
| If
B
is a set and
d
is an object, then the relationship |
|
d is an element of
B
we denote as
d
Î
B,
or |
| d not
Î
of B if d is not an element of B,
i.e., d
does not belong to the
set B. |
|
| Set
builder notation |
| A
set can be described by
specifying the properties which determines its elements, for
example |
| S
= {
n
Î
N |
3
≤
n ≤
8
} |
| denotes
the set S
of all natural numbers n
such that n
is in the range from 3 to
8 inclusive that is, |
| S
= {
3, 4, 5, 6, 7, 8
}. |
|
| Cardinal
number |
| A measure of the size of a set that does not take into account the order of its members, i.e., which only
specifies the total
number of its elements is called the cardinal number. |
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| Ordinal
number |
| A
measure of a set that takes account of the order as well as the
number of its elements is called the ordinal number. |
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| Equal
sets |
|
Two sets A
and B
are said to be equal if they have the same elements. This is written A
= B. |
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| Subsets |
| If set
S and T
are two sets such that every element of
S is also an
element of the set T,
then
S is said to be the subset
of the set T,
and is written as
S Í
T.
Every set is a subset of
itself. |
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| If
S is a subset of T,
but not
equal to T,
then
S is also proper subset
of T.
Therefore, a proper
subset is the one strictly contained within a larger
set, excluding some elements of the larger set. |
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| The
empty set or the null set |
| A
set which has no members is called the empty
set or the null set and is denoted by ø
or { }. |
| The
empty set is
therefore a
subset of every set. |
|
| Universal
set or universe |
| A universal set
U
is a set large enough to include all the elements of any set
relevant to a problem
under consideration.
Hence, the universal set is a set defined by the context. |
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| Power
set |
| A
power set is a set of which the elements are all the subsets of
a given set
S including the empty
set, written P
(S)
or 2S. |
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| Example: |
If
S
is the set { 1, 2, 3 } then power set of S,
P (S)
= { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}. |
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| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
