A = {1, 3, 2, 5}
n(A) = 4
Sets use “curly” brackets
The number of elements in Set A is 4
Sets are denoted by Capital letters
A3A7
3 is an element of A 7 is not an element of A
SETSA set is a collection of well defined distinct objects.The objects of the set are called
elements.
{1, 3, 2, 3, 5, 2} We never repeat elements in a set.{1, 2, 3, 5}
Sets can be represented by two methods:Rooster or Tabular form: In this the elements are separated by commas.e.g. set A of all odd natural numbers less than
10.A = {1,3,5,7,9}
Set builder method:In this the common property of the elements is
specified.e.g. set A of all odd natural numbers less than
10.A = {x : x is odd natural number less than 10}
Symbols Meaning
{ } enclose elements in set belongs to is a subset of (includes
equal sets) is a proper subset of is not a subset of is a superset of
Empty set: If a set doesn't contain any elements it is called
the empty set or the null set. It is denoted by or { }.
Singleton set: It is a set which contains only one element.
e.g. A = {0}
Finite set: It is a set which contains finite number of
different elements.e.g. A = {a,e,i,o,u}
Infinite set: It is a set which contains infinite number of
different elements.e.g. A = {x : x set of natural numbers}
Equal sets: If two or more sets contain the same elements,
they are called equal sets irrespective of the order.e.g. If A = {1,2,3} and B = {2,3,1} Then A = B
Number of Elements in Set
Possible Subsets Total Number of Possible Subsets
{a} {a} ; 2
{a , b} {a , b} ; {a} , {b} ,
4
{a , b , c} {a , b , c} , {a , b} , {a , c} , {b , c} , {a} , {b} , {c} ,
8
{a , b , c, d}{a , b , c , d} , {a , b, c} , {a , b , d} , {a , c , d} , {b , c , d} , {a , b} , {a , c} , {a , d} , {a , b} …… {D} ,
The number of possible subsets of a set of size n is 2n
16
The Power Set (P)The power set is the set of all subsets that can be created from a given set.Example:A = {a, b, c}P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, }
Cardinal Number of A SetIt is the number of elements in a set.Example:A = {a, b, c, d}n(A) = 4
A B
VENN DIAGRAMRepresentation of sets by means of diagrams known as:
Venn Diagrams are named after the English logician, John Venn. These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles.
A BA B
A BA B
This is the union symbol. It means the set that consists of all elements of set A and all elements of set B.
This is the intersect symbol. It means the set containing all elements that are in both A and B
A BA - B A B
A B
B- A
A B
Difference Of Sets
The difference of two sets A and B Is the set of elements which belongs to A but which do not belong to B. It is denoted by A – B. A – B = {x:x A and x B}∈ ∉ B – A = {x:x B and x A}∈ ∉
A BDisjoint Sets
ACompliment of a set
The two sets which do not have any elements in common are called disjoint sets.A B =
U
If U is the universal set, A is a subset, then compliment of A is A = {x:x U, x A}∈ ∉ or U - A
U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
A = {3,4,5,7,12,15}B = {2,3,4,6,7,10,14}A B = {3,7,4}A B = {2,3,4,5,6,7,10,12,14,15}A - B = {5,12,15}B – A = {2,6,10,14}
23
47
5
6
89
10
11
12
131415
A B
1
VENN DIAGRAM
U
A B
CA B C
Only A Only B
Only C
Only A B not C
Only B C not AOnly A C not B
Commutative Laws: A B = A B and A B = B AAssociative Laws: (A B) C = A (B C) and (A B) C = A (B C)Distributive Laws:A (B C) = (A B) (A C) and A (B C) = (A B) (A C)
Double Complement Law: (Ac)c = ADe Morgan’s Laws: (A B)c = Ac Bc and (A B)c = Ac Bc
100 people were surveyed. 52 people in a survey owned a cat. 36 people owned a dog. 24 did not own a dog or cat.
universal set is 100 people surveyed
C D
Set C is the cat owners and Set D is the dog owners. The sets are NOT disjoint. Some people could own both a dog and a cat.
24Since 24 did not own a dog or cat, there must be 76 that do own a cat or a dog.
52 + 36 = 88 so there must be 88 - 76 = 12 people that own both a dog and a cat.
1240 24
100
Example:
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