Summary Set theory l Fundamental concept and algebra of sets l BSc Maths

1. Fundamental Concepts
&
Algebra of Sets
Introductions
1. Definitions:
()Set: Acollection of distinctand distinguishable objects
of well defined
called set.The objects are known as
elements or members of a set.
is

e.g.,A= {1,2,3,a,b, c}
Here A is called set and 1,2,3, a, b, c are called elements of setA.
(i) Representations ofsets:There are two methods ofrepresenting a set
(a)Roster or Tabular form
(b) Set-builder form.
(a) In roster form, all the elements of a set are listed. the elements are
being separated by commas and are enclosed within curly brackets }. {
eg, The set ofall vowels in the English alphabet is {a, e, i, o, u}.
(b) In set-builder form all the elements of a set posseses a single common
property which is not possessed by any element outside the set.
e.g.,A= {x:x<8, where x isa natural number).
Note: There be no repeatition of elements in a set.
(ii)Null set (orvoid set or Empty set): The set having no element iscalled
null set. It is denoted by {}or ¢.
e.g., A= {x:risa collection oflifeon sun}
(iv)Singleton set: The set having only one elements is called singleton
set.

e.g,(a) A ={x:x is a collection of moon in the sky}
(b) B{6}
(v) Pair set: The set having
twoelements is called
pair set.
A =
eg..il {x:x*-9=0, xisinteger}
B
{13,16}
(vi) Finite set: The set having countable or finite number
of elements is
called finiteset.

e.g, A= {2, 3,4,5,6)
B {x:x is the collection ofgirls student in
(vii)Infinite set: The set
a
particular class}
having uncountable or infinitenumber of
iscalled elements
infiniteset.

e.g, A= {r:xis thecollection of stars in the sky}
B={1,2,3,4,.

, (viii) Sub sets: IfA and B are two sets such that every element ofB is a
an element ofA, then B is a subset of A.
It is denoted by BCA read as "
subset of" or, 'is contained in'.


e.g.,A={1,2,3, 4),B= {2,3}
BCA
(ix) Proper subset: IfA and B are two sets such that every element ofA
also anelement of B and at least one element of B that is not in A, then A is
proper subsetof B and we write AcB.
e.g., A= {a,b, c, d},B ={1,2, a, b, c, d} .. AcB.
(x)Power set: The collection ofall possible sub sets ofany given set A is
called power set. It is denoted by P(A)or24.
e.g A{1,3,6}
p(A)={{1} (3}, {6}, {1,3}.{3,6},{1,6}, {1,3,6).4}
(xi)Equality ofsets: Two setsA and B aresaid tobe equal ifevery element
of an element of B and also every element of B is an element ofA. It is
A is =B
written as A

c.g., itA= {x:r2-64 0},B =(-8,8}
Then A{-8,8}, b={-8,8} i.e.,A=B.
(xii) Universal
Set: If all the sets under consideration are subsets of a
fixed setthen this fixed set is called universal set.
It is denoted by Q or E or U or A.
e.g, A={1,2,5,6} Q={1,2,3,4, 5,6,7,8}
Here A C2.So 2 is a universal set.
(xiii) Equivalent set: Two
sets A and B areequivalent when their cardinal
numbers are equal
n(A)= n(B)
i.e.,


(xiv)Disjoint set:Two setsA and Bare said to be disjoint setsifthey have
no any elements in commen i.e., AnB= ¢.
eg, A= {a, b, c}, B= {d, e, f,g}
AnB= A &B are disjointsets.
(xv)Difference oftwo sets: The difference
of theset Arelative toB isaset
which contains only those elements of A which
does not belong to B.It is
denoted by B.A-
e.g A {a,b, c}, B= {b,c, d,e}
A-B= {a}
(xvi)Complement ofa set: The
complement ofaset A with
2 -A and denoted by A' orrespect
tothe
universal set Q is defined as the set
i.e. A'=Q-A A.
e.g.
A={3,4,5},2={1,2,3,4,5,6,7}
A=0-A= {1,2,6,7}
(xvii)Comparable set: The
B are saidto be comparable
setA and
ofthefollowing is satisfied:AcB ifeithe
orBcA.