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ADVANCED SET CONCEPTS
Jump right to the exercises!
To review basic properties of sets, click here.
To review interval and list notation for sets, click here.
SET-BUILDER NOTATION
Often, it is convenient to describe a set by stating a property that members of the set must satisfy.
In such cases, set-builder notation comes to the rescue.
Set-builder notation always has the following general form:
{
x |
a property that x must satisfy
}
The vertical bar, " | ", is read as such that
or with the property that .
For example,
{
x |
x>0}
is read as "the set of all x with the property that x is
greater than zero".
It can alternately be read as "the set of all x such that x is
greater than zero".
Of course, sets often have different names.
Here, the set could alternately be described using
interval notation:
{x |
x>0}
=(0,&Infinity;) .
Set-builder notation uses the concept of dummy variable.
Roughly, a dummy variable is just a name
given to something so that we have a way to talk about it;
the name used doesn't affect the result.
Dummy variables are used in function notation:
f(x)=
x+2 describes the rule "take a number and add two".
f(t)=
t+2 also describes the rule "take a number and add two".
In f(x)=
x+2 , the dummy variable is x.
In f(t)=
t+2 , the dummy variable is t.
The set {x |
x>0} can be written using the dummy variable t as
{t |
t>0} .
Or, it can be written using the dummy variable
w as
{w |
w>0} .
Of course, it's a good idea to stick to the normal conventions for naming variables:
for real numbers, use letters near the end of the alphabet; for integers, use letters near the middle of the alphabet.
CONNECTIVES FOR SETS
Addition, "+", is a connective for numbers. That is, two numbers x and
y can be connected to get a new number,
x+y.
Similarly, "∪" (union) and
"∩" (intersection) are connectives for sets; they combine two sets
to give a new set.
Let A and B be sets.
The set " A∪B " is read as " A union B ".
By definition,
A∪B
=
{x|x∈A
or
x∈B}
.
Thus, to find A∪B, you put in everything
from A, and also everything from B .
Venn diagrams are illustrations that are useful for showing the relationship between sets.
In the Venn diagram below, A is the left circle,
B is the right circle, and
A∪B is shaded.
The set " A∩B " is read as " A intersect B ".
By definition,
A∩B
=
{x|x∈A
and
x∈B}
.
Thus, A∩B consists of everything
that is common to both A and B; i.e.,
the overlap of A and B .
In the Venn diagram below, the double-hatched (darker) area in the middle is
A∩B .
Let U be a set (the universal set), and let
A be a subset of U .
The set A¯
is read as " A bar " or " the complement of A "
and is defined by
A¯
=
{x|x∈U
and
x∉A} .
Thus,
A¯
is everything that is not in A ; it is
everything that is outside of A .
In English, the word complement means to fill out or to make a whole,
and the set A¯
"fills out" the set
A to give all of U .
In the Venn diagram below, the universal set U
is the rectangle, the subset A is the circle,
and A¯
is the yellow region inside the rectangle, but
outside the circle.
EXAMPLE:
Let A={1
,2,3}
and
B={3
,4,5,6
} .
Then,
A∪B=
{1,2,3,
4,5,6}
and
A∩B=
{3} .
If U={1
,2,3,
4,5,6}
, then A¯
={4
,5,6} .
If U is the interval
[1,3
] , then
A¯
=(1,2
)∪(2,3
) .
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.