In mathematics it is useful to to place numbers with similar characteristics into sets. The following sets of numbers are used extensively in the study of algebra:

Integers	`{...,-3,-2,-1,0,1,2,3,...}`
Rational Numbers	`{`All terminating or repeating decimals`}`
Irrational Numbers	`{`All nonterminating, nonrepeating decimals`}`
Real Numbers	`{`All rational and irrational numbers`}`

If a number in decimal form terminates or repeats a block of digits, then the number is a rational number. Rational numbers can also be written in the form `p/q` where `p` and `q` are integers and `q\!=0`. For example,

`3/4=0.75` and `5/11=0.bar45`

are rational numbers. The bar over the 45 means that the block repeats without end; that is, `0.\bar45=0.4545454545...`.

In it's decimal form, an irrational number neither terminates nor repeats. For example, `0.272272227...` is a nonterminating, nonrepeating decimal and thus is an irrational number. One of the best-known irrational numbers is pi, denoted by the Greek symbol `pi`. The number `pi` is defined as the ratio of the circumference of a circle to its diameter. Often in applications, the rational number `3.14` or the rational number `22/7` is used as an approximation of the irrational number `pi`.

Each member of a set is called an element of the set. For instance, if `C={2,3,5}`, then the elements of `C` are `2,3,` and `5`. The notation `2inC` is read "`2` is an element of `C`." Set `A` is a subset of set `B` if every element of `A` is also an element of `B`, and we write `AsubeB`. For instance, the set of negative integers `{-1,-2,-3,-4,...}` is a subset of intergers. The set of positive intergers `{1,2,3,4,...}` (also known as natural numbers) is also a subset of the set of integers.

Prime numbers and composite numbers play an important role in almost every branch pf mathematics. A prime number is a positive integer other than `1` that has no positive-integer factors¹ other than itself and `1`. The ten smallest prime numbers are `2,3,5,7,11,13,17,19,23,` and `29`. Each of these numbers has only itself and `1` as factors. A composite number is a positive integer greater than `1` that is not a prime number. For example, `10` is a composite number because `10` has both `2` and `5` as factors. The ten smallest composite numbers are `4,6,8,9,10,12,14,15,16,` and `18`.

Sets are often written using set-builder notation, which makes use of a variable and a characteristic property that the elements of the set alone possess. This notation is especially useful to describe infinite sets. The set builder notation

`{x^2 | x` is an integer`}`

is read as "the set of all elements `x^2` such that `x` is an integer." This is the infinite set of perfect squares: `{0,1,4,9,16,25,36,49,...}`.
The empty set or null set is a set without any elements. The set of numbers that are both prime and also composite is an example of the null set. The null set is denoted by the symbol `O/`.

Just as addition and subtraction are operations performed on real numbers, there are operations performed on sets. Two of these set operations are called intersection and union. The intersection of sets `A` and `B`, denoted by `AnnB`, is the set of all elements belonging to both set `A` and set `B`. The union of sets `A` and `B`, denoted by `AuuB`, is the set of all elements belonging to set `A`, to set `B`, or to both.

Determine which of the following numbers are
a. integers	b. rational numbers	c. irrational numbers
d. real numbers	e. prime numbers	f. composite numbers
`-2 \ \ , \ \ 0 \ \ , \ \ 0.bar3 \ \ , \ \ pi \ \ , \ \ 6 \ \ , \ \ 7 \ \ , \ \ 41 \ \ , \ \ 51 \ \ , \ \ 0.71771777177771...`

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Example 2: Find the Intersection and the Union of Two Sets

Find each intersection or union, given `A={0,1,4,6,9}`, `B={1,3,5,7,9}`, and `P={x|x` is a prime number`<10}`.
a. `AnnB`	b. `AnnP`	c. `AuuB`	d. `AuuP`

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