*A set is a well defined collection of distinct objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[1]

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.[2]

Cantor's definition turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as anundefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic properties are that a set has elements, and that two sets are equal (one and the same) if and only if every element of one set is an element of the other.

*A set is a group of items (in most cases) of the same kind. The items in a set are called the elements or members of a set. A set is well-defined when all its members can be listed.

Examples of well-defined sets

1. A = the set of even numbers between 1 and 15= {even numbers between 1 and 15}= {2, 4, 6, 8, 10, 12, 14}

2. B = the set of multiples of 5 between 8 and 28= {multiples of 5 between 8 and 28}= {10, 15, 20, 25}

Note:

1. a set may be denoted by a capital letter (as shown above)2. {} means ‘the set of’

*Definition and terminology[edit]

Formally, a set S is called finite if there exists a bijection

for some natural number n. The number n is the set's cardinality, denoted as |S|. The empty set {} or Ø is considered finite, with cardinality zero.

If a set is finite, its elements may be written as a sequence:

In combinatorics, a finite set with n elements is sometimes called an n-set and a subset with k elements is called a k-subset. For example, the set {5,6,7} is a 3-set – a finite set with three elements – and {6,7} is a 2-subset of it.

*In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:

the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and

the set of all real numbers is an uncountably infinite set

*In mathematics, a singleton, also known as a unit set,[1] is a set with exactly one element. For example, the set {0} is a singleton.

The term is also used for a 1-tuple (a sequence with one element)

*In mathematics, and more specifically set theory, 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 assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set.

Null set was once a common synonym for "empty set", but is now a technical term in measure theory

*The roster method is one of two ways of representing the elements of a set using brackets, {}. For instance, all even numbers under ten would be represented:

{2,4,6,8}. The roster method is often associated with 'roster and rule' which is a way of finding a rule that the elements of a set follow. Sets can generally comprise any list of items.

*Definition of Rule

Rule is the procedure that a count must follow.

Solved Example on Rule

What is the rule for the table?

Choices:A. Add 2 and multiply by 1.B. Multiply by 2 and subtract 2.C. Subtract 2 and add 4.

D. Divide by 2 and add 3.

Correct Answer: DSolution: Step 1: Consider the first input value given in the table and try each rule on the value to check if we get the corresponding output value.

Step 2: Consider 4, (4 + 2) × 1 = 6, not 5. So, choice 1 does not work.Step 3: (4 × 2) - 2 = 6, not 5. So, choice 2 does not work.Step 4: (4 - 2) + 4 = 6, not 5. So, choice 3 does not work.Step 5: (4 ÷ 2) + 3 = 5, choice 4 works.Step 6: So, 'divide by 2 and add 3' is the rule for the table.

*Two sets are equal if they both have the same members.

ExampleIf, F = {20, 60, 80}And, G = {80, 60, 20}Then, F=G, that is both sets are equal.

Note: The order in which the members of a set are written does not matter.

*Two sets are equivalent if they have the same number of elements .

ExampleIf, F = {2, 4, 6, 8, 10}And, G = {10, 12, 18, 20, 22}Then, n(F)= n(G)= 5, that is, sets F and G are equivalent.

*Definition: Universal Set

A universal set does not have to be the set of everything that is known or thought to exist - such as the planets, extraterrestrial life, and the galaxies - even though that would be one example of a universal set. A universal set is all the elements, or members, of any group under consideration.

For instance, all the stars of the Milky Way galaxy could be a universal set if we are discussing all the stars of the Milky Way galaxy. This type of universal set might be appropriate for astronomers, but it is still a pretty large set of objects to consider.

*SUBSET

"Superset" redirects here. For other uses, see Superset (disambiguation).

Euler diagram showingA is a proper subset of Band conversely B is a proper superset of A

In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is asuperset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. Aand B may coincide. The relationship of one set being a subset of another is called inclusionor sometimes containment.

The subset relation defines a partial order on sets.

The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

*Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements , and therefore A has

a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers.[1] The cardinality of a set is also called its size, when no confusion with other notions of size[2] is possible.

The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolute value and the meaning depends on context.

Alternatively, the cardinality of a set A may be denoted by n(A), A , card(A), or # A.

*In mathematics, the power set (or powerset) of any set S, written ,P(S), ℙ(S), ℘(S) or 2S, is the set of all subsets of S, including the empty setand S itself.

In axiomatic set theory (as developed, for example, in the ZFCaxioms), the existence of the power set of any set is postulated by the axiom of power set.[1]

Any subset of is called a family of sets over S.

*two sets are said to be joint or overlapping sets ,if they have atleast one element in common e.g. A={3,4,5,6} B={1,4,7,8} are joint sets

*In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set.[1]For example, {1, 2, 3} and {4, 5, 6} are disjoint sets.

*In set theory, the union (denoted by ∪) of a collection of sets is the set of all distinct elementsin the collection.[1] It is one of the fundamental operations through which sets can be combined and related to each other.

Example of Union of Sets

If A = {1, 2, 3, 4, 5} and B = {2, 4, 6}, then the union of these sets is A ∪ B = {1, 2, 3, 4, 5, 6}.

*In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.[1]

For explanation of the symbols used in this article, refer to the table of mathematical symbols.

Example of Intersection of Sets

When Set A = {1, 2, 3, 7, 11, 13} and Set B = {1, 4, 7, 10, 13, 17},A ∩ B is all the common elements of the set A and B.Therefore, A ∩ B = {1, 7, 13}.

*In set theory, a complement of a set A refers to things not in (that is, things outside of) A. The relative complement of Awith respect to a set B is the set of elements

in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of all elements in U but not in A.