3.2 Sets Part 1:
3.2 Example 1:
In
topic 1.1 we have studied different type of numbers and let us try to
represent them using set notations. If N = {Set
of natural numbers}, W={Set of
whole Numbers}, Z= {Set
of integers} and Q= {set of rational numbers} Then N _{} W _{} Z _{} Q Note
All these are infinite sets. 

3.2 Example 2:
Let us consider the sets U = {Oxygen, Nitrogen, Hydrogen, Carbon dioxide, Sodium,
Carbon, Calcium} A = {Oxygen, Nitrogen, Hydrogen}
B = {Sodium, Carbon,
Calcium}
Note
that both A and B are subsets of U. Definition :
1) The ‘union’ (represented by the symbol
_{}) of two sets is defined as a set of all those elements
which are either in A or in B or in both A and B. A_{}B is pronounced as A union B. Thus in
this example A_{}B = {Oxygen,
Nitrogen, Hydrogen, Sodium, Carbon, Calcium}. In the
adjoining figure, the portion represented by grey colors is A_{}B. 2. The ‘intersection ’(represented
by the symbol _{}) of two sets is defined as a set of all those
elements which are present both in A and B.. The Symbol _{} is pronounced as
intersection. A_{}B is pronounced as A intersection B In this
example, note that sets A and B do not have any common element. When two sets
do not have common elements then they are called ‘disjoint’
sets. Note
that in this example A and B are disjoint sets as they do not have any common
element. Therefore A_{}B ={ }=_{}(Null set) The
representation of sets U, A and B as shown in the figure (in circular shape
or oval shape) is called representation by Venn diagram. 

In Venn diagram, the
universal set is represented by a rectangle
and sub sets are represented by circles or ovals.
3.2 Example 3 :
Let your class have a strength of 22. Assume that out
of these, 11 are members of the cricket team. Also assume that you have a
Hockey team of 11 players from these 22 students.It
is possible that few may be members of both the team and few may not be members
of any team.
The Venn diagram drawn on
the right helps us to find the players who are members of both the teams and
students who are not members of any team.
Let U
represent the set of students in your class. U = { X_{1},X_{2},X_{3}…………_{
}X_{22}) Let A
be the players representing cricket team. A = { X_{1},X_{3}, X_{4},X_{6,} X_{8},X_{11},X_{12},
X_{14},X_{17},X_{19},X_{21}) Let B
be the players representing hockey team. B = { X_{2},X_{3},X_{6}.X_{9},X_{10},X_{13},X_{14},X_{15},X_{18},X_{19},X_{20}} How do
we find players who represent both teams? How do you
find players who are not in any team? We find
that { X_{3},X_{6},X_{14},X_{19}} is the set
of four players who are in both
the teams. The set
A_{}B ={ X_{1},X_{2}, X_{3},X_{4,}
X_{6},X_{8}, X_{10},X_{11},X_{12},X_{13},X_{14},X_{15},X_{17},X_{18},
X_{19},X_{20,} X_{21}) is the set of players who are
in Cricket team or Hockey team. This is
the grey+ yellow+grey colored portion of the figure. We can
represent these sets using Venn diagrams as given on the right hand side. The
section shaded in yellow
color is the set of students who are in both the
teams and is represented by A_{}B. We also
find that {X_{5},X_{7},X_{9},X_{16},X_{22}}
is the set of five students who are
not in any team. 

3.2 Example 4 :
Let A = { 2,4,6,8}, Let B = { 2,4,6} _{} = { }
Observations: Since
in a set, the elements are written only once A_{}A = {2,4,6,8} = A Since _{} is a subset of every
set , A_{}_{} ={ 2,4,6,8} = A If B_{} A then B_{}A = {2, 4,6,8}= A IF B_{} A then A_{}B = {2,4,6} = B 

3.2 Problem 1: Draw the
Venn diagram for A_{}B_{}C and A_{}B_{}C if A = {1,5,7,9} and
B={1,3,7,10} and C= {5,6,7,8,9,10}
Solution :
Set A
is represented by the green colored circle. Set B is represented by
the blue colored circle. Set C
is represented by the red colored circle. (A_{}B)_{}C = ({1,3,5,7,9,10})_{}{5,6,7,8,9,10} = {1,3,5,6,7,8,910} (A_{}B)_{}C = ( { 1,7}_{}{5,6,7,8,9,10} = {7} 

3.2 Problem 2:
A = {x: x^{2}8x+12 =0} and B = {x:
x^{2}6x+8 =0} Find A_{}B and A_{}B
Solution :
We know
that x^{2}8x+12 = (x6)(x2). Thus x^{2}8x+12
= 0 is true when x=6 or x=2 We know
that x^{2}6x+8 = (x4)(x2). Thus x^{2}6x+8 = 0 is true when x=4 or x=2 Therefore
A = {6,2) (represented by blue colored circle) and B= {4,2} (represented by
green colored
circle). Hence A_{}B = {6, 4, 2}, A_{}B ={2} 

Let U = (Black, Pink, Brown, Purple, Violet,
Indigo, Blue, Green, Yellow, Orange, Red} A = {Violet, Indigo,
Blue, Green, Yellow, Orange, Red} The sets can be represented by a Venn diagram as
shown in the adjoining figure. Let us consider the set {Black, Pink, Brown,
Purple} What is special about this set? This set has
elements of U which are not in A. It is called ‘complement' of set A and is denoted
by A^{1}_{.} So A^{1}={Black, Pink, Brown, Purple} Definition :
A set is said to be ‘complement’
to another set U, if the elements in that set has elements which are in U but
not in itself . The complement of set A is denoted by A^{1}_{.} We notice that A^{1} _{} U and A _{} A^{1}=U and
A_{}A^{1}=_{}( The sets A and A^{1 } do not have any common element). (A^{1})^{1} = {elements of U
which are not in A^{1}} = {Violet, Indigo, Blue,
Green, Yellow, Orange, Red}= A 

3.2 Problem 3: If U = {Natural numbers less than 9}, A =
{Even numbers less than 9}, B = {Prime numbers less than 9}
Find A^{1}_{}B^{1} and A^{1}_{}B^{1}
Solution :
We have
U =
{1,2,3,4,5,6,7,8}, A = {2,4,6,8},
B = {2,3,5,7} A^{1}=
{ 1,3,5,7}( covered by blue colored quadrilateral ), B^{1}=
{ 1,4,6,8}( covered by red colored quadrilateral) A^{1}_{}B^{1} =
{1,3,4,5,6,7,8} A^{1}_{}B^{1}= {1} Let us find (A_{}B)^{1} and (A_{}B)^{1} A_{}B= {2,3,4,5,6,7,8}^{} (A_{}B)^{1}= {1} A_{}B = {2}( covered by orange colored square) (A_{}B)^{1}= { 1,3,4,5,6,7,8} What do
we notice? (A_{}B)^{1}= A^{1}_{}B^{1 } and (A_{}B)^{1}= A^{1}_{}B^{1} 

If A and
B are two sets then the ‘difference’ set
(AB) is defined as a set which has elements of A but not
of B.
3.2 Example 5 : Let H = {Squares of natural numbers less than
36} J = {1, and multiples of 2 less than 34} Find H_{}J HJ and JH
We have H = { 1,4,9,16,25}, J = {1,2,4,8,16,32} H_{}J = {1,4,16}(
covered by orange colored rectangle) H J =
{elements of H not in J} = {9,25}(covered
by red colored
rectangle) J  H =
{elements of J not in H} =
{2,8,32}(covered by blue colored rectangle) Notice
that HJ _{} JH Observations: For any
sets U and A (_{})^{1}=U and(U)^{1}= _{} AA= _{} 

3.2
Summary of learning
No 
Points
to remember 
1 
Definitions
of sets( 
2 
Venn
diagram 