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the power set of any set A is the set of all subsets of A, including the empty set and A itself, variously denoted as P(A)or P(S

For the set {a,b,c}:

The empty set {} is a subset of {a,b,c}

And these are subsets: {a}, {b} and {c}

And these are also subsets: {a,b}, {a,c} and {b,c}

And {a,b,c} is a subset of {a,b,c}

And altogether we get the Power Set of {a,b,c}:

P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }

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A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets[2] (i.e., X is a disjoint union of the subsets).

Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and:

1. The union of the elements of P is equal to X. (The elements of P are said to cover X.)

2. The intersection of any two distinct elements of P is empty. (We say the elements of P are pairwise disjoint.)

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The Cartesian product A × B between two sets A and B is the set of all possible ordered pairs with first element from A and second element from B. For example, if A = {x, y} and           B = {3, 6, 9},
then A × B = {(x, 3), (x, 6), (x, 9), (y, 3), (y, 6), (y, 9)}.

A×B={(x,y):x∈A and y∈B}.

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In set theory, the complement of a set A refers to elements 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 elements in U but not in A.

The relative complement of A with respect to a set B, also termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A.

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If E denotes the set of all positive even numbers and O denotes the set of all positive odd numbers, then their union yields the entire set of positive integers, and their intersection is the empty set. Any two sets whose intersection is the empty set are said to be disjoint.

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The intersection operation is denoted by the symbol ∩.
The set A ∩ B you read “A intersection B” or “the intersection of A and B” is defined as the set composed of all elements that belong to both A and B.

A ∩ B = {x : x ∈ A and x ∈ B}

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We write A-B and read A minus B.
In symbols A-B {x|x A and x ₵ B}.

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The symbol ∪ is employed to denote the union of two sets. The union of two sets is the set containing all of the elements from both of those sets. The set A ∪ B you read “A union B” or “the union of A and B”.

A ∪ B = {x : x ∈ A or x ∈ B}

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We say the difference between two sets A and B, considered in the order,the set of elements of A that do not belong to B.

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Definition 1: If two sets A and B have same cardinality, if there exists a objective function from set A to B.

Definition 2: Two sets A and B are said to be equivalent if they have the same cardinality i.e. n(A) = n(B).

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To be equivalent, the sets should have the same cardinality. This means that there should be one to one correspondence between elements of both the sets. Here, one to one correspondence means that for each element in the set A, there exists an element in the set B till the sets get exhausted.

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If the condition discussed above is not met, then the sets are said to be unequal. This is represented by

A ≠ B

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Two sets A and B can be equal only if each element of set A is also the element of the set B. Also, if two sets are the subsets of each other, they are said to be equal. This is represented by:

A = B

A ⊂ B and B ⊂ A ⟺ A = B

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Two sets can either be equivalent, equal or unequal to each other.

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A={2, 3, 4, 5, 6, 7, 8} is proper subset of B={1, 2, 3, 4, 5, 6, 7, 8}.

If A is a subset of B, then B is called the super set of A. The symbol ⊂ is called the inclusion symbol. If A is not a subset of B, we write A⊄B. By the definition of a subset, it is clear that the empty set and the set A itself are always subsets of A. These two subsets are called the improper subsets of A.

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Set of all points in a line segment is an infinite set.

. N = {1, 2, 3, ……….} i.e. set of all natural numbers is an infinite set.

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A set is said to be an infinite set whose elements cannot be listed if it has an unlimited (i.e. uncountable) by the natural number 1, 2, 3, 4, ………… n, for any natural number n is called a infinite set.

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1. Let P = {5, 10, 15, 20, 25, 30}

Then, P is a finite set and n(P) = 6.

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The element does not occur more than once in a set.

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The number of distinct elements counted in a finite set S is denoted by n(S). The number of elements of a finite set A is called the order or cardinal number of a set A and is symbolically denoted by n(A).

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A set is said to be a finite set if it is either void set or the process of counting of elements surely comes to an end is called a finite set. In a finite set the element can be listed if it has a limited i.e. countable by natural number 1, 2, 3, ……… and the process of listing terminates at a certain natural number N.

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1. Odd numbers which are divisible by 2 are well defined objects for an empty set.
2. The positive integers which are less than -1 are also well defined for en empty set.

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The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. The intersection of two disjoint sets (two sets that contain no elements in common) is the null set

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A={x:x€N^x<_5}
B={x:x€E^0<y<_50}

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A= Set of frist five natural numbers.
B= Set of positive odd integers less than or equal to fifty.

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A set is usualy denoted by capital letters,A,B,C... and the elements are denoted by samall letters, a,b,c,d...

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A set is usualy denoted by capital letters,A,B,C... and the elements are denoted by samall letters, a,b,c,d...

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A set is a collection of objects that are called elements of the set

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A={1,2,3,4,5};B={1,3,5,7,9...}...