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}
. It is
sometimes said that “A is contained in B” or even “B contains A”. 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 an element of 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 }.
The symbol
[a,b]
where a and b are real numbers, 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.
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
A ∖ B ≡ {x ∈ A : x ∕∈ B }.
Set notation is used whenever convenient.
To illustrate the use of this notation relative to intervals consider three examples of inequalities. Their
solutions will be written in the notation just described.
Example 1.1.1Solve the inequality 2x + 4 ≤ x − 8
x ≤−12 is the answer. This is written in terms of an interval as (−∞,−12].
Example 1.1.2Solve the inequality
(x+ 1)
(2x − 3)
≥ 0.
The solution is x ≤−1 or x ≥
3
2
. In terms of set notation this is denoted by (−∞,−1] ∪ [
3
2
,∞).
Example 1.1.3Solve the inequality x
(x +2)
≥−4.
This is true for any value of x. It is written as ℝ or