defines the set of rational numbers. The vertical bar  is read "such that."
This definition works
because there is an understanding that
the set contains every
p
/
q
that satisfies the condition to the
right
of the bar.
Another example, the set {1, 3, 5, 7, 9, 11, .
.. } can be defined as:
{ n
∈
Z

n = 2k+1
for some integer k >= 0 }
Some other important facts about sets:
1) Two sets are equal
if and only if they have the same elements:
{1, 3, 5, 5, 3, 1, 1 } = {5, 1, 3}
2) If A and B are sets, then A is a
subset
of B if and only if, every element of A is also an
element of B:
A = {1,2,5,7} B = {1, 5}
C = {1, 5} B is a subset of A; B is a subset of C.
Another way of defining set equality is to say A = B if and only if A is a subset of B and B is
a subset of A.
Note that if you need to prove set equality, you have to prove two
things.
3) If A is a subset of B but A does not equal B, then A is called a
proper subset
.
In the
example above, B is a proper subset of A, but B is not a proper subset of C.
4)
Venn diagrams
are often used to graphically represent sets.
In Venn diagrams, the
Universal Set (U) contains all the objects under consideration.
This is represented by a
rectangle.
Inside the rectangle are circles or other geometrical figures used to represent
sets.
Venn diagrams are often used to illustrate the relationship between sets.
For example,
the first diagram below shows that A is a subset of B.