# Class 9 – Set Theory

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## Online Tests

Topic Sub Topic Online Practice Test
Set Theory
• Review of set theory
• Venn diagrams
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Set Theory
• Cardinality of sets
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## 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} 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 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}

=

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.

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}

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

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}

Element wise properties of sets:

i)

ii)    _

iii)  _

iv)  _

v)     _

vi)   _

vii)  _

viii)  _

ix)   _

x)    _

1.

2.

3.

4.    If A and B are disjoint sets then