Set Theory
A set is a collection of objects, or
elements
. Typically, the type
of all the elements in a set is the same. (For example, all the
elements in a set could be integers.) However, it is possible to
have different types of elements in a set. (An analogy for this is
that usually a bookbag contains just books. But sometimes it
may contain other elements such as pencils and folders as
well.)
We have two standard methods of denoting the elements in a
set:
1) Explicitly list the elements inside of a set of curly braces({}),
as follows: {1, 2, 3, 4}
2) Give a description of the elements in a set inside of a set of
curly braces as follows: { 2x  x
∈
N }.
In order to understand the second method, we must define the
various symbols that are used in this notation. Here is a list of
the symbols we will be using:
  translates to “such that”.
∈
 “is an element of”
⊂
 “is a subset of” (Note: in the book they define this symbol
as “is a proper subset of”, and use
⊆
to mean “is a subset of”.)
Now we have to define what a subset is. A subset is also a set.
So, if we have sets A and B, A
⊂
B if for all x
∈
A, x
∈
B.
In layman’s terms, a set A is a subset of a set B, if all the
elements in the set A also lie in the set B.
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We still have to define what { 2x  x
∈
N } really means. Here it
is in English:
“The set of all numbers of the form 2x such that x is an element
of the natural numbers.” (Note: The set N denotes the natural
numbers, or the nonnegative integers, according to the book.)
So, the set above could also be listed as {0, 2, 4, 6, ...}
Now that we have gotten that out of the way, let’s talk about
the empty set(
∅
). The empty set is a set with no elements in it.
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 Spring '09
 Sets, Natural number

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