FORMULAS AND FACTS TO REMEMBER FOR MATH 121 EXAM 2
SET THEORY
•
Set
: A collection of objects where order does not matter and where repeated elements do
not count.
•
Subset
: B is a subset of A if all the elements of B are contained in A. Therefore, if A=B,
then B is still a subset of A.
•
Empty set
: Set with no elements. Denoted as:
∅
or
{ }
. The empty set is always
a subset
of any set.
•
Equivalent sets
: Denoted as “
∼
.” A
∼
B if they have the same number of
distinct
elements.
(e.g.
{
a, b, c
} ∼ {
1, 2, 3
}
).
•
Equal sets
: A=B if they have the exact same elements. (e.g.
{
1, 2, 3
}
=
{
3, 1, 2
}
).
•
Disjoint sets
: A and B are disjoint if they do not have any elements in common. Denoted
as: A
∩
B =
∅
. (e.g. if A =
{
a, c
}
and B =
{
d, f, g
}
, A
∩
B =
∅
).
•
Set builder notation
: Describing a set in terms of some characteristic that all the elements
share. (e.g. A =
{
1, 3, 5, 7, 9
}
can be rewritten in set builder notation as
{
x

x is an odd
number between 0 and 10
}
)
•
Roster notation
: Describing a set by listing all the elements in the set. (e.g. A =
{
New
England states
}
can be rewritten in roster notation as A =
{
NH, VT, MA, CT, RI, ME
}
).
•
Universal set
: Denoted as “U.” Set of all objects under consideration (e.g. if A =
{
a, b,
c
}
, B =
{
d, f
}
and A and B are subsets of U, a suitable U could be
{
letters of the English
alphabet
}
).
• ∈
= “belongs to” (e.g.
a
∈ {
letters of the alphabet
}
)
•
/
∈
= “does not belong to” (e.g. 3
/
∈ {
even numbers
}
)
• 
= “such that” (e.g.
{
x

x
∈
Real #s
}
= “x such that x is a real number”)
•
Intersection of 2 sets
: Denoted as A
∩
B. The set of all elements that are in A
and
B. (e.g.
if A=
{
a, b, c, d
}
and B =
{
c, d, f, g
}
, A
∩
B =
{
c, d
}
).
•
Union of 2 sets
: Denoted as A
∪
B. The set of all elements that are in A
or
B. (e.g. if A
=
{
a, b, c, d
}
and B =
{
c, d, f, g
}
, A
∪
B =
{
a, b, c, d, f, g
}
).
•
Complement of a set
: Denoted as A
0
. The set of all elements that are in U but
not
in A,
i.e.
{
x

x
∈
U and x
/
∈
A
}
(e.g. if U =
{
a, b, c, d
}
and A =
{
b, c
}
, A
0
=
{
a, d
}
).
•
A