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SETS
A set is a well-defined collection of objects or numbers.
The objects that make up a set are called its elements or members.
One way to write a set is to place the elements inside a pair of braces {
} and to
separate the elements with commas. In a set, the order of the elements does not matter,
but elements are not repeated. If a set contains no elements it is said to be empty. We
use the symbols  or   to represent an empty set.
To specify a set, you must identify its elements.
Frequently you can specify a set by listing or making a roster of its elements.
For example, the set of vowels of the English alphabet can be represented in
roster form by {a,e,i,o,u}.
A second way of specifying a set is by giving a rule or condition enabling you to
decide whether or not an object belongs to the set. For example, the set {0, 1, 2,
3} can be represented by the rule { the whole numbers less than 4}. You may
think of this method as providing a word description of a set.
A third way of specifying a set is by graph. For example, the set { 5, 4, 3,
2 } can be represented by the graph
6 5 4 3 2 1
0
Two sets are said to be equal if they have the same elements. The sets {5, 3, 2} and {2,
5, 3} are equal, but {5, 3, 2} and {5, 3, 1} are not equal
Two sets that contain the same number of elements are said to be equivalent. The sets
{a, e, i, o, u} and {1, 2, 3, 4, 5} are equivalent because they have the same number of
elements.
Since equivalent sets have the same number of elements, we can pair up the elements of
the sets exactly. Therefore, we say that the elements of equivalent sets have a one-to-one
correspondence. Conversely, if there is a one-to-one correspondence between the
elements of two sets, the sets are equivalent.
{ a, e, i,
o,
u}
{ 1,
2,
3,
4,
5}
The diagram above illustrates the one-to-one correspondence between the elements of the
equivalent sets {a, e, i, o, u} and {1, 2, 3, 4, 5}.
A set is said to be finite if its elements can be counted and the counting comes to an end.
Very large sets are finite if you can at least imagine the counting process coming to an
end.
The empty set is finite because it is possible to count all the elements: none.
A set is said to be infinite if it is not possible to count the elements in the set.
SETS OF NUMBERS
Following is a list of important sets of numbers and their names. All Algebra II students
must know these familiar sets.

Natural Numbers (also known as “counting numbers” or “positive integers”)
{1, 2, 3, 4, 5, …}

Whole Numbers (also known as “non-negative integers”)
{0, 1, 2, 3, 4, 5, …}

Integers
{…, -3, -2, -1, 0, 1, 2, 3, …}

Negative Integers
{…, -5, -4, -3, -2, -1 }
SUBSETS
Let A = { 2, 4, 6 }
Let B = { 2, 4, 6, 8, 10 }
Clearly, every element of A is also an element of B.
Whenever each element of a set A is also an element of a set B, we say, “ A is a subset of
B ”. Using math symbols we write A  B .

Every set is a subset of itself.
For example, since every integer is a member of the set of integers, it is certainly true
that the set of integers is a subset of itself. We write {Integers}  {Integers}.
(Note: we use the symbol  when we wish to communicate the fact that a set is a
subset of itself as in {A}  {A}

By agreement, the empty set is accepted as a subset of every set.
For example, in the case of set A above, { }  { A } or we may write,   { A
}
Since the empty set is a subset of itself we write   
INTERSECTION AND UNION OF SETS
The intersection of two or more sets is the set of elements the sets have in common.
The symbol for intersection is .
The union of two or more sets is the set of all the elements in all of the sets, without
repetition.
The symbol for union is .
Example for Practice Set 4
Ex(1) A = {3, 2, 1, 0 } and B = {-1, 0, 1, 2, 3}
Ans. A  B = { 1, 0 }
A  B = {3, 2, 1, 0, 1, 2, 3}
Give A  B and A  B.