sets

chapter-sets

what is a set?

a set is a group of similar objects written as one .

it can be any object like

days of a week, months of a year , all the animals , etc.

overlapping sets-

overlapping sets are sets which have at least one element in common . if there are two sets A and B , then A and B are said to be overlapping if A∩B≠ ∅ 

FOR EXAMPLE- if there are two sets

A={1 , 2 , 3 , 4}

B={1 , 2 , 3 }

then , A and B are said to be overlapping sets because some elements which are in B are in A

disjoint set-

disjoint sets is the exact reverse of overlapping sets , i.e. , sets which have no elements in common are called disjoint sets. therefore we can say that, if there are two sets A and B , then A and B are said to be disjoint if A∩B = ∅ 

FOR EXAMPLE- if there are two sets

A={1 , 2 , 3 , 4}

B={5 , 6 , 8 }

then , A and B are said to be disjoint sets because no elements are in common in set A and B.

signs of sets-

• ∪(union)

• ∩(intersection)

• subsets-⊆(subset) , ⊂(proper subset)

• supersets - ⊇(superset) , ⊃(proper superset) 

• (If there are two sets A and B)

•  A - B(A minus B) 

• A' (complement)

SET SIGN-

union -∪

union sign is used to denote the elements which are in any two sets but any element should not be repeated.

FOR EXAMPLE- if there are two sets-

A ={1 , 2 , 3 , 4}

B ={1 , 2 , 3 }

A∪B={1 , 2 , 3 , 4}

because the elements{1 , 2 , 3 , 4} are present in set 'A' and set 'B' and since the elements{1 , 2 , 3} are repeated they are only written once.

SET SIGN-

intersection-∩

intersection sign is used to denote the elements which are same in any two sets.

FOR EXAMPLE- if there are two sets-

A ={1 , 2 , 3 , 4}

B ={1 , 2 , 3 }

A∩B={1 , 2 , 3 }

because the elements{1 , 2 , 3 } are present both in set 'A' and set 'B'.

SET SIGN-

subset-⊆ &

superset-⊇

subset and superset denote that every element of any two sets are same . they act as (=) sign . (NOTE- the side towards which the mouth opens can have more elements)

FOR EXAMPLE- if there are two sets

A ={1 , 2 , 3 , 4}

B ={1 , 2 , 3 , 4}

A⊆B = true

it means that both sets A and B have the same element and B can have more elements

A⊇B = true

it means that both sets A and B have the same element and A can have more elements

subset

SET SIGN-

proper subset-⊂&

proper superset - ⊃

proper superset and proper subset are used to denote that (if there are two sets A and B)

1.) A ⊂ B - set A has all elements that are in set B , but B has more elements.

2.)A ⊃ B- set A has all elements as set B , but A has more elements

proper superset

SET SIGN-

minus - '-'

minus is used to remove all the elements in a set from another set

FOR EXAMPLE- if there are two sets-

A ={1 , 2 , 3 , 4}

B ={1 , 2 , 3 }

A-B={4 }

the elements {1 , 2 , 3} are removed from elements{1 , 2 , 3 , 4} , that is equal to {4}

UNIVERSAL SET-

universal sets are those sets which contain all elements of a given type , from which all other sets will derive their elements . it is denoted by 'U' or phi 'ξ'

FOR EXAMPLE-

ξ={months of a year}

={jan, feb , march , april , may , june , july , sep , oct , nov , dec}

A={may , june , july}

here, elements of A are the same as universal set but universal set has more elements.

SET SIGN-

complement- '

complement sign is used to denote the elements which are not the in A but in the universal set

so,

A'={jan, feb , march , april , sep , oct , nov , dec}

here , elements of A {may , june , july} are not present in set A'

cardinal number-

cardinal number of a set is the number of elements in a set

FOR EXAMPLE- if there is a set

A={1 , 2 , 3 }

then the cardinal number of A=3 because the number of elements in the set are 3.

cardinal number is represented by n(set_name) , i.e. , here , cardinal number of set A =n(A)

formulas for finding cardinal numbers-

• n(A∪B) = n(A)+n(B)-n(A∩B)

• n(A-B) = n(A∪B) - n(B) = n(A)-n(A∩B)

• n(B-A) = n(A∪B) - n(A) = n(B) - n(A∩B)

• n(A∪B) = n(A-B)+n(B-A) +n(A∩B)

• n(ξ)=n(A)+n(A')

Venn diagram-

venn diagram is diagram which is used to represent sets including universal sets. with this diagram we can find union and intersection also along with complements.

the current diagram shows AuB

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