Set theory is important mainly because it serves as a foundation for the rest of mathematics–it provides the axioms from which the rest of mathematics is built up.

Axioms are important to get right, because all of mathematics rests on them. If there are too few axioms, you can prove very little and mathematics would not be very interesting. If there are too many axioms, you can prove almost anything, and mathematics would also not be interesting.

A function in set theory world is simply a mapping of some (or all) elements from Set A to some (or all) elements in Set B. In the example above, the collection of all the possible elements in A is known as the domain; while the elements in A that act as inputs are specially named arguments.

The empty set is a subset of any set. This is because we form subsets of a set X by selecting (or not selecting) elements from X. One option for a subset is to use no elements at all from X. This gives us the empty set.

There are three main ways to identify a set:

• A written description,
• List or Roster method,
• Set builder Notation,

(2) Set – builder method or Rule method : In this method, a set is described by a characterizing property P(x) of its elements x. In such a case the set is described by {x : P(x) holds} or {x | P(x) holds}, which is read as ‘the set of all x such that P(x) holds’. The symbol ‘|’ or ‘:’ is read as ‘such that’.

The roster method is defined as a way to show the elements of a set by listing the elements inside of brackets. An example of the roster method is to write the set of numbers from 1 to 10 as {1,2,3,4,5,6,7,8,9 and 10}. An example of the roster method is to write the seasons as {summer, fall, winter and spring}.

symbol ∅

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced.

There is a special set that has no elements. This set is called the empty set, or null set, and is denoted by ∅ (or Ø, or 0) or {}. It is often helpful to have a visual representation of sets.

1 subset

If an equation has no solutions, its solution set is the empty set or null set–a set with no members, denoted Ø. For example, the solution set to x2 = – 9 is Ø, because no number, when squared, is equal to a negative number. Sometimes we will be given a set of values from which to find a solution–a replacement set.

Any Set that does not contain any element is called the empty or null or void set. The symbol used to represent an empty set is – {} or φ. Examples: Let A = {x : 9 < x < 10, x is a natural number} will be a null set because there is NO natural number between numbers 9 and 10.

There are many types of set in the set theory:

• Singleton set. If a set contains only one element it is called to be a singleton set.
• Finite Set.
• Infinite set.
• Equal set.
• Null set/ empty set.
• Subset.
• Proper set.
• Improper set.

A singleton set is a set containing exactly one element. For example, {a}, {∅}, and { {a} } are all singleton sets (the lone member of { {a} } is {a}). The cardinality or size of a set is the number of elements it contains.

Sets that have precisely the same elements. They don’t have to be in the same order. Example: {1,2,3,4} and {3,4,2,1} are equal. Note: it must work both ways, each element of the 1st set must be in the 2nd set and each element of the 2nd set must be in the 1st set.

The definition of equal is someone or something with the same quantity or value, or someone having the same rights as another. An example of equal is one cup being the same as eight ounces. An example of equal is women getting the same pay as men for the same work.

A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set….Some commonly used sets are as follows:

1. N: Set of all natural numbers.
2. Z: Set of all integers.
3. Q: Set of all rational numbers.
4. R: Set of all real numbers.
5. Z+: Set of all positive integers.

Types of a Set

• Finite Set. A set which contains a definite number of elements is called a finite set.
• Infinite Set. A set which contains infinite number of elements is called an infinite set.
• Subset.
• Proper Subset.
• Universal Set.
• Empty Set or Null Set.
• Singleton Set or Unit Set.
• Equal Set.

Example: The set {a, b, c} has eight subsets. They are: ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, and {a, b, c}. Proper Subset: A proper subset is a special type of subset. There are two requirements for set A to be a proper subset of set B.