Sets
•
Sets are unordered collections of distinct objects.
•
Sets can be defined or specified in many ways:
–
By explicitly enumerating their members or elements
e.g.
S = { a, b, c}
Note: If S' = { b, c, a}, then S and S' denote the same set (that is, S' = S)
–
By specifying a condition for membership
S =
{ x
∈
∆

P(x) }, reads "S is the set of all x in
such that P(x) is true"
P is called a "predicate" ( a function from set
to {true, false} )
E.g.
S = { x
UCF  x is a CS major }
•
The
empty set
is denoted,
Φ
, and is the set with no members; that is,
Φ
= { }.
Also, the predicate,
x
∈
Φ
, is always false
!
•
If S
≠
Φ
, then there exists an x for which x
∈
S is true; this predicate is read
"
x is an element of
S
" or "
x is a member of
S
".
The symbol
"
∈
" denotes the
member relation
.
•
If
A and B are sets,
then we write "
A
⊆
B
" to mean that
A is a subset of
B
.
This means that for all x
∈
A, x
∈
B.
Or,
2200
x [x
∈
A
⇒
x
∈
B].
•
The expression, "
A
⊂
B
" means that
A is a proper subset of
B
.
Mathematically,
2200
x [x
∈
A
⇒
x
∈
B] and
5
y [ y
∈
B and y
∉
A]
•
The
cross product of two sets A and B
is denoted,
A
×
B
, and is the set
defined as follows:
A
×
B = { (a,b) 
a
∈
A and b
∈
B }
.
"(a,b)" is an
expression composed from elements, a,b, selected arbitrarily from sets A and
B, respectively.
If A
≠
B, then A
×
B
≠
B
×
A.
In fact,
Exercise:
Prove A
×
B = B
×
A if and only if A = B.