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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