2 – SETS, FUNCTIONS AND GROUPS 1

Definitions:
SET
A set is an aggregate, class or collection of distinct objects or numbers
EQUAL SETS
Two sets A and B are said to be equal if and only if the have same elements and denoted by A =
B. e.g. if A = {1,2,3} and B = {3,2,1} then A and B are equal.
EQUIVALENT SETS
Two sets A and B are said to be equivalent if and only if the number of elements in A = number of
elements in B, where A ≠ B. e.g. if A = {a,b,c} and B = {1,2,3} then A is equivalent to B but A ≠ B.
FINITE AND INFINITE SETS
A set P is said to be finite if the number of elements is finite and infinite if the number of elements
is infinte. e.g. N = {1,2,3,…………….} is infinite while P = {1,2,3,4,………….10000000} is finite.
EMPTY SET OR NULL SET
If a set A has no element then it is called empty set. It is denoted by {} or Ø.
SUBSET
If every element of set A is also the element of set B, then A is called the subset of B denoted by
A B. e.g. if A = {a, b, c} and B = {a, b, c, d, e} then A B.
POWER SET
The power set of a set “S” is the set of all possible subsets of “S” denoted by P(S). if n is the total
number of elements of S then its power set will contain 2n elements.
UNIVERSAL SET
It is the set of all the members under consideration denoted by “U”.
COMPLEMENTARY SET
Le t U be universal set and A is the subset of U, the U – A is called the compliment of A denoted
by A’.
OPERATIONS ON SETS
Union
Let A and B be two sets, their union denoted by AUB is the set which contains all the
elements of both the sets i.e. AUB = {x/x ЄA or xЄB}. e.g. if A = {1,2,3} and B = {3,4,5}
then AUB = {1,2,3,4,5}.
Intersection
The intersection of two sets A and B denoted by A∩B is the set which contain the
common elements of A and B i.e. A∩B = {x/x Є A and x Є B}. e.g. if A = {1,2,3} and B =
{2,3,4,5} then A∩B = {2,3}.

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar

 2 – SETS, FUNCTIONS AND GROUPS 2

Difference of two sets

Difference of two sets A and B denoted by A – B or A\B is the set of elements which are
in A but not in B i.e. A – B = {x/x Є A but x }. e.g. if A = {1,2,3} and B = {2,3,4,5,6}
then A – B = {1} and B – A = {4,5,6}. We see that A – B ≠ B – A.

PROPERTIES OF OPERATIONS ON SETS

Commutative property of union and intersection
AUB=BUA
A∩B=B∩A
Associative property of Union and intersection
AU(BUC) = (AUB)UC
A∩(B∩C) = (A∩B)∩C
Distributive property of U over ∩
AU(B∩C) = (AUB) ∩(AUC)
Distributive property of ∩ over U
A∩(BUC) = (A∩B)U(A∩C)
VENN DIAGRAM
Venn was the 1st English mathematician who represented sets and their operations by means of
diagram. e.g.

These diagrams are for sets A and B which are not disjoint. For disjoint sets we make circles
separately. The rectangle in all the above figures represents the universal set U.

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar

 2 – SETS, FUNCTIONS AND GROUPS 3

SIMPLE PROPOSITIONAL LOGIC

A proposition is a statement which is either true or false. e.g.
“It is raining”. This statement is either true or false so this is a proposition denoted by p,q,r,s,t… or
A,B,C,D…...
we write “1” for true proposition and “0” for false.
NEGATION OF A PROPOSITION
Let p = “it is raining” is a proposition, its negation is denoted by “~p” is given by:
~p = it is not raining.
CONNECTIVE SYMBELS
~, are called connective symbols, where
~ stands for negation
V stands for or
۸ stands for and
When two or more propositions are connected by means of connective symbols, it is called
compound or complex proposition.e.g. AvB, A ۸ B are compound propositions.
TRUTH TABLES
A B AvB
A ~A A B A^ B
1 0 1 1 1
1 1 1
0 1 1 0 1
1 0 0
0 1 1
0 1 0
0 0 0
0 0 0

Where T[AvB]= Minimum of [T(A), T(B)]

And T[A^B] = Maximum of [T(A), T(B)]
Also ~A = 1 – T(A)
CONDITIONAL PROPOSITIONS
The proposition of the form “if x = 2 then x2 = 4” is called conditional proposition.i.e. “if A then B”
or “A implies B” or “AB” is called conditional proposition.
EQUIVALENT COMPOUND PROPOSITONS
Two compound propositions are said to be equivalent if their truth tables are identical.
A  B is equivalent to ~AvB.
A  B is equivalent to ~B  ~A.
B  A is equivalent to ~A  ~B.
The converse of A  B is B  A.
The Inverse of A  B is ~A  ~B.
The Contra positive of A  B is ~B  ~A.

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar

 2 – SETS, FUNCTIONS AND GROUPS 4

BINARY RELATION
Let A and B be two sets, then any subset of AXB is called a binary relation from A to B.e.g. if A =
{1,2} and B = {3,4} then AXB = {(1,3), (1,4),(2,3), (2,4)}.
Now there are 4 elements in AXB, the number of possible subsets of AXB are 24 = 16. therefore
16 possible binary relations can be drawn from A to B.
DOMAIN OF BINARY RELATION
The 1st element of each ordered pair in a binary relation is called the domain of that binary
relation.e.g. if R = {(1,3),(2,4)}, then its domain is {1,2}.
RANGE OF BINARY RELATION
The 2nd element of each ordered pair in a binary relation is called the range of that binary
relation.e.g. in the above set R, the range of R is {3,4}.
FUNCTION
Let A and B be two non-empty sets and R be a binary operation from A to B, then R will be a
function from A into B if:
(i) Domain of R = A
(ii) Every element of A is related to exactly one and unique element of B.
Condition (ii) above can also be stated as:
“The 1st co – ordinate of ordered pairs in R do not repeat.”
The functions are usually denoted by f, g, h etc.
TYPES OF FUCTIONS
(i) Linear function: the function f:x ax + b is called linear function.
(ii) Quadratic function: the function f:x ax2 + bx + c is called quadratic function.
(iii) Square root function: the function f:x  is called square root function.
(iv) One to one function (1 – 1 function): If distinct elements have distinct images, it is
called 1 – 1 function.e.g.

a x

b y

c z

A B
(v) Onto function: Let f:AB be a function, if range of f = B, then f is called onto function.
(vi) Bijective function: if a function is both one to one and onto, then it is called Bijective
function.

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar

 2 – SETS, FUNCTIONS AND GROUPS 5

BINARY OPERATION
Let “S” be a non – empty set. A function from SXS into S called a binary operation in S. to
understand this we will construct a multiplication table.
Let S = {1, - 1}, then multiplication table is:
. 1 -1

1 1 -1

-1 -1 1
After multiplying the elements of set “S” we get nothing new but the elements of “S”. therefore
multiplication is binary operation on set “S”.
GROUPOID
Let “S” be a non – empty set and is a binary operation defined on it, then the ordered pair
(S, ) is called a Groupoid.
SEMI GROUP
A groupoid (S, ) is called a semi group if “ ” is associative.
MONOID
A semi group (S, ) is called a Monoid if it contains the identity element w.r.t. .
GROUP
Let “S” be a non – empty set and “ ” is defined on “S”, then S is called a group if:
(i) “S” is closed under “ ”.
(ii) “ ” is associative.
(iii) There exist an identity element in “S” w.r.t. “ ”.
(iv) Each element in “S” has the inverse in “S” w.r.t. “ ”.

Prepared by: Inayat Khan Lecturer in Mathematics Edwardes College Peshawar