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 