Sets in Mathmatics

SET:

Collection of well defined objects is called set.

For example, the set of digits consists of the collection of numbers 0,1,2,3,4,5,6,,7,8,9.

If we use the symbol A to denote the set of the digits, than we can write

A={,0,1,2,3,4,5,6,7,8,9}

The braces {  } are used to enclose elements in the set. This method of denoting a set is called the roster method.

A second way to denote a set is to use set-builder notation, the set ”A”  of digits is written as

A = {xǀx is a digit}

“A” is the set of all x such that x is a digit.

Using Set-builder Notation and the Roster Method

       (S)                       E={XǀX is an even digit} ={0,2,4,6,8}

       (T)                       O={XǀX is an odd digit}={ 1,3,5,7,9}

       (U)                      N={XǀX is natural numbers} ={I,2,3,4,…}

       (V)                      W={xǀx is whole numbers}={0,1,2,3,4,…}

The elements of a set, we do not list an element more than once because the elements of a set are unique. Also, order in which the elements are listed is not relevant.

Thus, e.g , {4,5} and {5,4} both represent the same set.

SUBSET:

If every element of a set ” A” is also an element of a set “B”, then we say that “A”  is a subset of “B” . e.g A={2,3,4} and B={2,3,4,5,}

Every set is subset of itself. e.g  A ⊆ A, B ⊆ B.

Empty is subset of every set. Empty set is denoted by { } or Ø

SUPER SET:

If A is subset of B  then B is superset of A, it can be write as  B ⊇ A

If two sets A and B  have the same elements, then A equals B. e.g B={2,3,4} and B={3,4,2}.

PROPER  SUBSET:

If A and B are two sets , than A is called proper subset of B, if  A ⊆ B but B ⊇ A i.e., A ≠B  we denoted by  A⊂B

e.g A={5,6,7,8}

n(A)=4

B={5,6,7,,8,9}

n(B)=5

All the elements of A are present in B but the element “9 “ of set B is not present in set A. Then we say that A is proper subset of B, symbolically it writes as A⊂B .

Sher Afzal Ranais

Author & Editor

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