Class 9 – Set Theory

Take practice tests in Set Theory Review

laptop_img Online Tests

Topic Sub Topic Online Practice Test
Set Theory
  • Review of set theory
  • Venn diagrams
Take Test See More Questions
Set Theory
  • Cardinality of sets
Take Test See More Questions

file_img Study Material

Introduction: In mathematics set theory is a branch that deals with the properties of well defined collection of objects, which may or may not be of a mathematical nature such as numbers or literals. The present synopsis, gives us the review on set theory and also recalls operations on sets.

 

Union of set: Let A and B are two sets then union of A and B denoted by  is defined as a set containing elements present either in A or in B or in both A and B.

Example:
Let A = { 1,4,6,8,9 }, B = { 2,3,4,11,13 }

= { 1,4,6,8,9 }  { 2,3,4,11,13 }

         = {1, 2, 3, 4, 6, 8, 9, 11, 13}

http://images.clipartpanda.com/bloc-clipart-note-md.png

1)    .

2)    To list elements of  we first write all elements of A and then write the elements of B which are not present in A.

 

Example:
Let A = {1, 3, 5}; B = {1, 2, 3, 5, 7, 9}

AB = {1, 3, 5}  {1, 2, 3, 5, 7, 9}

         = {1, 2, 3, 5, 7, 9} = B

http://images.clipartpanda.com/bloc-clipart-note-md.png If AB then = B

 

Some Properties on the Operations of Union:

 

Intersection of Sets: Let A and B are two sets. The intersection of A and B denoted by is defined as the set containing elements present in both A and B.

 

Example:

Let A = { is a factor of 32}

      B = { is a factor of 48}

Then A = { 1, 2, 4, 8, 16, 32 },

             B = { 1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

Then  = { 1, 2, 4, 8, 16, 32 }
{ 1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

= {1, 2, 4, 8,16}

http://images.clipartpanda.com/bloc-clipart-note-md.png =

i) If ABthen AB =A,

ii) =,

iii)

Some Properties on Operations of Intersection:

i)                                           (Commutativelaw).

ii)                   (Associativelaw).

iii)                   (Lawof  and )

iv)                                               (Idempotentlaw)

v)       

vi)  

 

Disjoint Sets: Two sets are said to be disjoint if they have no elements in common.

Example: P= Set of all odd natural numbers

               Q= Set of all even natural numbers

Then P = {1, 3, 5, 7, 9…………………….}

        Q = {2, 4, 6, 8, 10…………………..}

Clearly P and Q do not have any common elements. Thus P and Q are disjoint sets.

http://images.clipartpanda.com/bloc-clipart-note-md.png If A and Bare disjoint sets then

 

Over Lapping sets: Two sets A and B are said to be overlapping, if they have at least one element common to them.

Example: Let P = {a, b, c, d, e}

                  Q= {a, e, i, o, u}

Here P and Q have two elements a, ein common. Thus P and Q are overlapping sets.

 

Complement of a set: Let m be the universal set and let A.Then, complement of A denoted by or  is defined as the set of elements of  whichare not present in A.

Examples:

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

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

2) Let  ={ x/x Î N, 50 < x < 60}

        A = {x/x is a pirme 50 < x < 60}

Here = {51,52, 53, 54, 55, 56, 57, 58, 59 }

        A = {53, 59}

= – A= {51, 52, 54, 55, 56, 57, 58}

http://images.clipartpanda.com/bloc-clipart-note-md.png 

 

Some properties on complement of sets:

1.Complement laws: (i)  (ii)

2.De Morgan’s law: (i)  (ii)

3. Law of double complementation:

4.Laws of empty set and universal set:   and.

5.

6.

 

Difference of sets: Let A and B be the two non-empty sets.

i)A – B is defined as the set containing elements in A but not in B.

ii)B – A is defined as set containing elements in B but not in A.

Example: Let A = {1,3, 4, 6, 8} B = { 2, 4, 6, 8, 11, 13}

Then A – B = {1, 3, 4, 6, 8} – {2, 4, 6, 8, 11, 13} = {1, 3}

B- A = {2, 4, 6, 8, 11, 13} – {1, 3, 4, 6, 8} = {2, 11, 13}

A – B  B- A

http://images.clipartpanda.com/bloc-clipart-note-md.png

1.    If  then,

2.    If A and B are disjoint sets then
i) A – B  = A and         ii) B – A = B

    

Symmetric difference of twosets:  If A and B are two sets. Then symmetric difference of two sets denoted by  isdefined as

Example: Let A = {1, 4, 6, 8, 9, 10}; B = {2, 3,4, 6, 8, 11, 13}

A – B = {1, 4, 6, 8, 9, 10} – {2, 3, 4,6, 8, 11, 13}

         ={1, 9, 10}

B – A = {2, 3, 4, 6, 8, 11, 13} – {1, 4,6, 8, 9, 10}

         ={2, 3, 11, 13}

       

       = {1, 2, 3, 9, 10, 11, 13}

http://images.clipartpanda.com/bloc-clipart-note-md.png 

 

Element wise properties of sets:

i)     

ii)    _

iii)  _

iv)  _

v)     _

vi)   _

vii)  _

viii)  _

ix)   _

x)    _

http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gif

1.   

2.   

3.   

4.    If A and B are disjoint sets then