Set Vocabulary
With the idea of sets in place, we can continue by looking at more vocabulary related to sets. In the last section, we said that elements, or members, of a set A are said to belong to A. In other words, an object a of a set A is denoted: a 
A. Object a is NOT an element of set A is expressed by: a 
A.
I. Set Builder Notation
In Example 1.3, we gave the rule for the set {California, Connecticut, Colorado} as {States in the US that begin with the letter C}. Another way to write this set is:
II. Subsets
The set of {a, b, c, d, e} contains another set, namely {a, b, c}. It also contains the set {c, d}. These two sets are called subsets and they are not the only subsets of {a, b, c, d, e}. Note that two sets A and B are equal if and only if their elements are identical.
III. Two Special Sets
There are two important special sets. THe first is the empty set, denoted
or { }, which contains no elements. The second is the universal set, denoted 
, which contains all elements. In other words, every set is considered a subset of .
IV. Cardinality
The cardinality of a set is the number of elements it contains. If a set S has a finite number of elements, S is the finite set with n elements. We denote cardinality of S by |S|.
V. Cartesian Products
The cartesian product is the set of all ordered pairs generated by the elements of A and B.
Example 2.1 If A = {1,3} and B = {a, b, c}, what is the cartesian product of A and B? B and A?
Solution 2.1 The Cartesian Product of A and B,
Example 2.2 What is A x B x C, where A = {0,1}, B={1,2} and C={0,1,2}?
Solution 2.1 The Cartesian Product of A, B and C is
A. Object a is NOT an element of set A is expressed by: a A.
I. Set Builder Notation
In Example 1.3, we gave the rule for the set {California, Connecticut, Colorado} as {States in the US that begin with the letter C}. Another way to write this set is:
{x:x is a state beginning with the letter C}
This type of notation is called set builder notation and it is the preferred way of describing sets. Another example is:{x:x is even, positive, and greater than 20}
In general:| Set builder notation has the form {x:x has a property P} and is read "the set of x such that x has property P. |
II. Subsets
The set of {a, b, c, d, e} contains another set, namely {a, b, c}. It also contains the set {c, d}. These two sets are called subsets and they are not the only subsets of {a, b, c, d, e}. Note that two sets A and B are equal if and only if their elements are identical.
| Two sets A and B are equal if and only if for all x A, x B.
|
Set A is a subset of B, denoted A  B, if every element of A is an element of B.If A B and A  B, then A is a proper subset of B and is denoted A  B.
|
III. Two Special Sets
There are two important special sets. THe first is the empty set, denoted
or { }, which contains no elements. The second is the universal set, denoted , which contains all elements. In other words, every set is considered a subset of .
IV. Cardinality
The cardinality of a set is the number of elements it contains. If a set S has a finite number of elements, S is the finite set with n elements. We denote cardinality of S by |S|.
V. Cartesian Products
The cartesian product is the set of all ordered pairs generated by the elements of A and B.
| Let A and B be sets. The Cartesian Product of A and B, denoted by A x B, is the set of all ordered pairs (a, b) where a A and b B. Hence A x B = {(a, b) | a |
b Example 2.1 If A = {1,3} and B = {a, b, c}, what is the cartesian product of A and B? B and A?
Solution 2.1 The Cartesian Product of A and B,
A x B = {(1,a), (1,b), (1,c), (3,a), (3,b), (3,c)}
B x A = {(a,1), (a,3), (b,1), (b,3), (c,1), (c,3)}
Example 2.2 What is A x B x C, where A = {0,1}, B={1,2} and C={0,1,2}?
Solution 2.1 The Cartesian Product of A, B and C is
A x B x C = {(0,1,0), (0,1,1), (0,1,2), (0,2,0), (0,2,1), (0,2,2), (1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)}