3. SETS
A SET is a well-defined collection of objects.
Theory of sets was developed by George Cantor
• We use { ……} (curly brackets) for presentation of set elements.
• € = element Eg. A = { 2,3,4,5} so 2 € A
Examples of sets :
1) Rivers of India
2) Various kinds of triangles
3) Sets of all odd numbers
4) Sets of vowels of English alphabets
4. TYPES OF SETS
1) Sigleton set is a set with only one element Eg. A = { 2 }
2) Empty (NULL) set is the set with no element which is denoted by symbol ø
Cardinality :
A measure of a set’s size that is the number of elements in the set.
• Example :
• A = { 1,2,3,4,5,6}
• Cardinality of set A = 6 or n(A) = 6 OR |A| = 6
• Cardinality Rule for intersection of union of set :
• Let A and B are two finite sets.
1) If n(A ∪ B) = n(A)+n(B)-n(B ∩ A)
if A ∩ B =ø
2) If n (A ∩ B) =n(A) + n(B)
5. TYPES OF SETS
3) Finite set is a set with finite cardinality (number)
Eg. B = { 1, 2, 3, ………,49,50) (cardinality of B) n(B) = 50 or |B|= 50
4) An Infinite set is a set with infinite cardinality
Eg. 1) E = { 1,2,3,4,5,……………} ( cardinality of E) n(E)= ∞ or |E|= ∞
5) SUBSET - A is a Subset of B if every element in A is present is also in B.
We also say that B is a SUPERSET of A. eg. A = {1,2,3} B = {1,2,3,4,5,6}
So A is a subset of B OR A ⊂ B and every ø (empty) set is a subset of every set
6) UNIVERSAL SET – set which has elements of all the related sets, without any
repetition of elements. Symbol U
A = { 1,2,3} B= { 1,a,b,c}
Universal Set ( U) = { 1,2,3, a,b,c}
6. UNION OF SETS
Let A and B of any two sets. The union of A and B is the set which consists of all the
elements of A and all the elements of B, the common elements taken only once.
Example :
1)Let A = {1, 2,3,4} B= {3,4,5,6}
we have A ∪ B = {1,2,3,4,5,6}
Properties of Union of sets :
1) A ∪ B = B ∪ A ( commutative law)
2) A ∪ A = A (idempotent law)
3) A ∪ ø (empty set) = A ( law of identity element )
4) A ∪ ( B ∪ C) = (A ∪ B) ∪ C ( associative law )
7. INTERSECTION OF SETS
• The intersection of sets A and B is the set of all elements which are common to both
A and B
• Example :
• Let A ={ 1,2, 3,4} B ={3,4,5,6}
• A ∩ B = {3,4}
• PROPERTIES :
• 1) A ∩ B = B ∩ A (commutative law)
• 2) A ∩ A = A (idempotent law)
• 3) A ∩ ᴓ = ᴓ (law of empty)
• 4) A ∩ ( B ∩ C) = (A ∩ B ) ∩ C (associative law)
• 5) A ∩ ( B ∪ C ) = ( NA ∩ B) ∪ (A ∩ B)
8. COMPLEMENT OF SETS
Let U be the universal set and A a subset of U then the complement of A is the set of all elements
of U which are not the elements of A. Symbol A'.
Eg. A={1,2,3,4} U={1,2,3,4,5,6,7,8}
COMPLEMENT OF A OR A' = {5,6,7,8}
The complement of the union of two sets is the intersection of their complements and the
complement of the intersection of two sets is the union of their complements thses are called as
De Morgan's law.
(A ∪ B)'=A' ∩ B' ( A ∩ B)'=A'∪ B'
Properties :
1) COMPLEMENT LAW =A ∪ A'=U AND A ∩ A'= ø
2) LAW OF DOUBLE COMPLEMENTATION ( A')'=A
3) LAWS OF EMPTY SET AND UNIVERSAL SET
U'=ø AND ø' = U
∩
