Class 9 – Set Theory
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Set Theory 

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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: = { 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: 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, 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 nonempty 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