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Introduction to Discrete Mathematics Set Theory
In discrete mathematics, the set theory are the branch of mathematics that learned about the sets, which are the collections of objects. Even though any type of objects can be collected into a set, set theory is applied most often to objects that are related to mathematics. Now we will see the examples of discrete mathematics set theory.


The natural numbers
The 'counting' numbers (or whole numbers) starting at 1, are called the natural numbers. This set is sometimes denoted by N. So N = {1, 2, 3, ...}

Integers
All whole numbers, positive, negative and zero form the set of integers. It is sometimes denoted by Z. So Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Real numbers
If we expand the set of integers to include all decimal numbers, we form the set of real numbers. The set of reals is sometimes denoted by R.
e.g. 8.127127127...

Rational numbers
Those real numbers whose decimal digits are finite in number, or which recur, are called rational numbers. The set of rationals is sometimes denoted by the letter Q.
For example: 0.5, -17, 2/17, 82.01, 3.282828..

Irrational numbers
If a number can't be represented exactly by a fraction p/q, it is said to be irrational.

•  Examples include: √2, √3, π.

Universal Set
The set of all the 'things' currently under discussion is called the universal set (or sometimes, simply the universe). It is denoted by U.

Empty set
The set containing no elements at all is called the null set, or empty set. It is denoted by a pair of empty braces: {} or by the symbol .

Equality
Two sets A and B are said to be equal if and only if they have exactly the same elements. In this case, we simply write:
A = B

Subsets
If all the elements of a set A are also elements of a set B, then we say that A is a subset of B, and we write:
A ⊆ B

Disjoint
Two sets are said to be disjoint if they have no elements in common. For example:

• If A = {even numbers} and B = {1, 3, 5, 11, 19}, then A and B are disjoint.

Intersection
Region iii, where the two loops overlap (the region corresponding to 'Y' followed by 'Y'), is called the intersection of the sets A and B. It is denoted by A ∩ B. So we can define intersection as follows:

• The intersection of two sets A and B, written A ∩ B, is the set of elements that are in A and in B.

(Note that in symbolic logic, a similar symbol, , is used to connect two logical propositions with the AND operator.)

Union
In a similar way we can define the union of two sets as follows:

•       The union of two sets A and B, written A ∪ B, is the set of elements that are in A or in B (or both).

The union, then, is represented by regions ii, iii and iv in Fig. 7.
(Again, in logic a similar symbol, , is used to connect two propositions with the OR operator.)

Difference

•       The difference of two sets A and B (also known as the set-theoretic difference of A and B, or the  relative complement of B in A) is the set of elements that are in A but not inB.

This is written A - B, or sometimes A \ B.

Complement
So far, we have considered operations in which two sets combine to form a third: binary operations. Now we look at a unary operation - one that involves just one set.

• The set of elements that are not in a set A is called the complement of A. It is written A′ (or sometimes AC, or ).

Cardinality

• Finally, in this section on Set Operations we look at an operation on a set that yields not another set, but an integer.

The cardinality of a finite set A, written | A | (sometimes #(A) or n(A)), is the number of (distinct) elements in A. So, for example:
If A = {lower case letters of the alphabet}, | A | = 26.

A∪B

A∩B

 
A –B

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