1.3 Set Notation
A set is just a collection of things called elements. Often these are also referred to as
points in calculus. For example
would be a set consisting of the elements 1,2,3,
and 8. To indicate that 3 is an element of
it is customary to write
means 9 is not an element of
Sometimes a rule
specifies a set. For example you could specify a set as all integers larger than 2.
would be written as S
This notation says: the set of all integers x,
such that x >
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,
is a subset of
The same statement about the two sets may also
be written as
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,
because these numbers are those which are in at
least one of the two sets.
Note that 3 is in both of these sets.
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
because 3 and 8 are those elements the two
sets have in common. In general,
When with real numbers,
denotes the set of real numbers,
a ≤ x ≤ b
) denotes the set of real numbers such that a ≤ x < b.
consists of the set of real numbers,
such that a < x < b
] indicates the
set of numbers, x
such that a < x ≤ b.
) means the set of all numbers,
such that x ≥ 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
are called endpoints of the interval.
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
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
the above description, I have used the usual description of this set in terms of
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.
Set notation is used whenever convenient.