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KIET Group of Institutions, Ghaziabad: Department of Computer Applications

The document defines basic set operations including union, intersection, difference, and complement of sets. It provides examples to illustrate each operation. Laws of algebra are also described for sets under union, intersection, and complement operations. These laws include idempotent, associative, commutative, distributive, De Morgan's, identity, complement, and involution laws.

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49 views4 pages

KIET Group of Institutions, Ghaziabad: Department of Computer Applications

The document defines basic set operations including union, intersection, difference, and complement of sets. It provides examples to illustrate each operation. Laws of algebra are also described for sets under union, intersection, and complement operations. These laws include idempotent, associative, commutative, distributive, De Morgan's, identity, complement, and involution laws.

Uploaded by

neemarawat11
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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KIET Group of Institutions, Ghaziabad

Department of Computer Applications


(An ISO – 9001: 2015 Certified & ‘A’ Grade accredited Institution by NAAC)

Operations on Sets

The basic set operations are:

1. Union of Sets: Union of Sets A and B is defined to be the set of all those elements which belong
to A or B or both and is denoted by A∪B.

1. A∪B = {x: x ∈ A or x ∈ B}  

Example: Let A = {1, 2, 3},       B= {3, 4, 5, 6}


A∪B = {1, 2, 3, 4, 5, 6}.

2. Intersection of Sets: Intersection of two sets A and B is the set of all those elements which belong
to both A and B and is denoted by A ∩ B.

1. A ∩ B = {x: x ∈ A and x ∈ B}  

Example: Let A = {11, 12, 13},       B = {13, 14, 15}


A ∩ B = {13}.

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KIET Group of Institutions, Ghaziabad
Department of Computer Applications
(An ISO – 9001: 2015 Certified & ‘A’ Grade accredited Institution by NAAC)

3. Difference of Sets: The difference of two sets A and B is a set of all those elements which belongs
to A but do not belong to B and is denoted by A - B.

1. A - B = {x: x ∈ A and x ∉ B}  

Example: Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A - B = {3, 4} and B - A = {5, 6}

4. Complement of a Set: The Complement of a Set A is a set of all those elements of the universal
set which do not belong to A and is denoted by Ac.

Ac = U - A = {x: x ∈ U and x ∉ A} = {x: x ∉ A}

Example: Let U is the set of all natural numbers.


A = {1, 2, 3}
Ac = {all natural numbers except 1, 2, and 3}.

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KIET Group of Institutions, Ghaziabad
Department of Computer Applications
(An ISO – 9001: 2015 Certified & ‘A’ Grade accredited Institution by NAAC)

5. Symmetric Difference of Sets: The symmetric difference of two sets A and B is the set
containing all the elements that are in A or B but not in both and is denoted by A ⨁ B i.e.

1. A ⨁ B = (A ∪ B) - (A ∩ B)  

Example: Let A = {a, b, c, d}
B = {a, b, l, m}
A ⨁ B = {c, d, l, m}

Algebra of Sets

Sets under the operations of union, intersection, and complement satisfy various laws (identities)
which are listed in below Table.

Table: Law of Algebra of Sets

Idempotent Laws (a) A ∪ A = A (b) A ∩ A = A

Associative Laws (a) (A ∪ B) ∪ C = A ∪ (B ∪ (b) (A ∩ B) ∩ C = A ∩ (B ∩


C) C)

Commutative (a) A ∪ B = B ∪ A (b) A ∩ B = B ∩ A


Laws

Distributive Laws (a) A ∪ (B ∩ C) = (A ∪ B) ∩ (b) A ∩ (B ∪ C) =(A ∩ B) ∪


(A ∪ C) (A ∩ C)

De Morgan's (a) (A ∪B)c=Ac∩ Bc (b) (A ∩B)c=Ac∪ Bc


Laws

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KIET Group of Institutions, Ghaziabad
Department of Computer Applications
(An ISO – 9001: 2015 Certified & ‘A’ Grade accredited Institution by NAAC)

Identity Laws (a) A ∪ ∅ = A (c) A ∩ U =A


(b) A ∪ U = U (d) A ∩ ∅ = ∅

Complement (a) A ∪ Ac= U (c) Uc= ∅


Laws (b) A ∩ Ac= ∅ (d) ∅c = U

Involution Law (a) (Ac)c = A

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