A set is just a collection of things called elements. Often these are also referred to as points in
calculus. For example
{1,2,3,8}
would be a set consisting of the elements 1,2,3, and 8. To
indicate that 3 is an element of
{1,2,3,8}
, it is customary to write 3 ∈
{1,2,3,8}
.
9
∕∈
{1,2,3,8}
means 9 is not an element of
{1,2,3,8}
. Sometimes a rule specifies a set. For
example you could specify a set as all integers larger than 2. This would be written
as S =
{x ∈ ℤ : x > 2}
. This notation says: the set of all integers, x, such that
x > 2.
If A and B are sets with the property that every element of A is an element of B, then A
is a subset of B. For example,
{1,2,3,8}
is a subset of
{1,2,3,4,5,8}
, in symbols,
{1,2,3,8}
⊆
{1,2,3,4,5,8}
. The same statement about the two sets may also be written as
{1,2,3,4,5,8}
⊇
{1,2,3,8}
.
The union of two sets is the set consisting of everything which is contained
in at least one of the sets, A or B. As an example of the union of two sets,
{1,2,3,8}
∪
{3,4,7,8}
=
{1,2,3,4,7,8}
because these numbers are those which are in at
least one of the two sets.In general
A ∪ B ≡ {x : x ∈ A or x ∈ B }.
Be sure you understand that something which is in both A and B is in the union. It is not an
exclusive or.
The intersection of two sets, A and B consists of everything which is in both of the sets.
Thus
{1,2,3,8}
∩
{3,4,7,8}
=
{3,8}
because 3 and 8 are those elements the two sets have in
common. In general,
A ∩ B ≡ {x : x ∈ A and x ∈ B }.
When with real numbers,
[a,b]
denotes the set of real numbers, x, such that a ≤ x ≤ b and
[a,b) denotes the set of real numbers such that a ≤ x < b.
(a,b)
consists of the set of real
numbers, x such that a < x < b and (a,b] indicates the set of numbers, x such that
a < x ≤ b. [a,∞) means the set of all numbers, x such that x ≥ a and (−∞,a] means
the set of all real numbers which are less than or equal to a. These sorts of sets
of real numbers are called intervals. The two points, a and b are called endpoints
of the interval. Other intervals such as
(− ∞, b)
are defined by analogy to what
was just explained. In general, the curved parenthesis indicates the end point it
sits next to is not included while the square parenthesis indicates this end point is
included. The reason that there will always be a curved parenthesis next to ∞ or
−∞ is that these are not real numbers. Therefore, they cannot be included in any
set of real numbers. It is assumed that the reader is already familiar with order
which is discussed in the next section more carefully. The emphasis here is on the
geometric significance of these intervals. That is [a,b) consists of all points of the
number line which are to the right of a possibly equaling a and to the left of b.
In the above description, I have used the usual description of this set in terms of
order.
A special set which needs to be given a name is the empty set also called the null set,
denoted by ∅. Thus ∅ is defined as the set which has no elements in it. Mathematicians like to
say the empty set is a subset of every set. The reason they say this is that if it were not so,
there would have to exist a set, A, such that ∅ has something in it which is not in A. However,
∅ has nothing in it and so the least intellectual discomfort is achieved by saying
∅⊆ A.
If A and B are two sets, A ∖ B denotes the set of things which are in A but not in B.
Thus