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Set (mathematics)


A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics.



A set is a gathering together into a whole of definite, distinct objects of our perception and of our thought - which are called elements of the set.

The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.


Basic operations

There are several fundamental operations for constructing new sets from given sets.




The union of A and B, denoted A  B

Two sets can be "added" together. The union of A and B, denoted by A  B, is the set of all things which are members of either A or B.


1.  {1, 2} {red, white} ={1, 2, red, white}.

2.  {1, 2, green} {red, white, green} ={1, 2, red, white, green}.

3.  {1, 2} {1, 2} = {1, 2}.

(Our solved example in mathguru.com uses this concept)

Some basic properties of unions:

1.  A  B = B  A.

2.  A  (B  C) = (A  B)  C.

3.  A  (A  B).

4.  A  A = A.

5.  A  A.

6.  A  B if and only if A  B = B.



A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A  B, is the set of all things which are members of both A and B. If A  B = , then A and B are said to be disjoint.


The intersection of A and B, denoted A  B.


1.  {1, 2} {red, white} = .

2.  {1, 2, green} {red, white, green} = {green}.

3.  {1, 2} {1, 2} = {1, 2}.

(Our solved example in mathguru.com uses this concept)


Some basic properties of intersections:

  • A  B = B  A.
  • A  (B  C) = (A  B)  C.
  • A  B  A.
  • A  A = A.
  • A  = .

       A  B if and only if A  B = A.




The relative complement
of B in A


The complement of A in U


The symmetric difference of A and B

Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (orA − B), is the set of all elements which are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.


1.  {1, 2} \ {red, white} = {1, 2}.

2.  {1, 2, green} \ {red, white, green} = {1, 2}.

3.  {1, 2} \ {1, 2} = .

4.  {1, 2, 3, 4} \ {1, 3} = {2, 4}.

5.  If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then E′ = O.

(Our solved example in mathguru.com uses this concept)


Some basic properties of complements:

1.  A \ B ≠ B \ A.

2.  A  A′ = U.

3.  A  A′ = .

4.  (A′)′ = A.

5.  A \ A = .

6.  U′ =  and ′ = U.

7.  A \ B = A  B′.



The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.