Download A binary operation

KENDRIYA VIDYALAYA
IFFCO,GANDHIDHAM
MATHS PROJECT
ON
BINARY OPERATIONS
 Name:
Swetha,Sumouly,Prachi,Sejal.
 Standard: XII
 Subject: Mathematics
 Topic: Binary Operations
 Teacher-In-charge: Geetha Ma’am
 School: KV,IFFCO Gandhidham.
 Year: 2010-2011
We hereby extend our sincere gratitude to our
teacher/mentor Madam Geetha for alloting such
an interesting project,to our Principal Mrs.Sangeeta
Gutain , for her co-operation and guidance,to our
parents for their help,concern and blessings.We
gratefully acknowledge the valuable and precious
contributions and support of the above mentioned
people in making this project a SUCCESS………..
SR.NO.
TOPIC
01
Introduction (Binary Operations)
SLIDE
NO.
05
02
Commutative Binary Operation
07
03
Associative Binary Operation
09
o4
Identity on Binary Operation
11
05
Invertible Binary Operation
13
06
07
Bibliography
Remarks & Suggestions
15
16

BINARY OPERATION:
A binary operation * on a set A is a
function from AXA→A. We denote it by
*(a,b)=a*b.+,-,*,/,these operations
between two operands result in binary
operation.
E.g. A binary operation on ‘R’
+:R×R→R is given by (a,b)→a+b
E.g. A binary operation of division is not
possible on ‘R’ because ‘R’ includes 0 also and
in ‘a/b’ form if ‘b’ is zero then product
becomes not defined.

COMMUTATIVE OPERATION
A binary operation * on the set x is called
commutative if a*b=b*a for all a,b ε X.
E.g. 5+2=2+5, where a=5,b=2 and * is
operation (+), this satisfies the condition
of a+b=b+a,i.e 7=7.
E.g. Show that * is a function from RXR→R
defined by a*b=a+2b is not commutative?
Solution:
a*b=a+2b (given)
b*a=b+2a,
but it is clear that a*b ≠ b*a , that means
a+2b ≠ b+2a.
let us take a=2 and b=3
a+2b= 2+2(3)=8,but
b+2a=3+2(2)=7,
it is clear that a+2b ≠ b+2a as 8 ≠ 7.
Hence proved…

ASSOCIATIVE OPERATION:
A binary operation * : AXA →A is said to
be associative if (a*b)*c=a*(b*c) for all
a,b,c ε A.
E.g. (8+5)+2=8+(5+2),a=8,b=5 & c=2 and *
is operation(+),this satisfies the condition
(a*b)*c=a*(b*c) implies 15=15.
E.g. Show that * is a function from RXR→R
defined by a*b=a+2b is not associative?
Solution:
(a*b)*c=(a+2b)*c (given)
a*(b*c)=a*(b+2c)
but it is clear that (a*b)*c ≠ a*(b*c)
that means
(a+2b)+c ≠ b+2a.
let us take a=8 ,b=5 and c=3
(a+2b)*c= (8+2(5))*3=24, but
a+(b+2c)=8*(5+2(3))=30
it is clear that (a+2b)*c ≠ a*(b+2c) as 24 ≠ 30.
Hence proved…

IDENTITY OPERATION:
A binary operation * : AXA →A ,an element eεA
if it exists, is called identity for operation *,if
a*e=a=e*a for all aεA.

ADDITIVE IDENTITY:
zero is identity for the addition operation.

MULTIPLICATIVE IDENTITY:
one is identity for the multiplication operation.
E.g. Show that zero and one are additive and
multiplicative identity on R. But there is no
identity element for subtraction and division.
Solution:
a+0=0+a= a, shows 0 is additive identity and
ax1=1xa= a , shows 1 is multiplicative identity.
but
a-e=e-a= a , there is no such value for e.
a/e=e/a=a , there is no such value for e.
Hence proved…

INVERTIBLE BINARY OPERATION:
A binary operation * : AXA →A ,with the
identity element in A , an element a ε A is
said to be invertible with respect to the
operation*, if there exists an element b in A
such that a*b=e=b*a and b is called the
inverse of a and is denoted by a-1.
E.g. Show that –a and 1/a(a!=0) are the
inverse of a for addition(+) and
multiplication operation(x) on R.
Solution:
As a+(-a)=a-a=0 & (-a)+a=0,shows –a is the
inverse of a for addition.
Similarly, for a!=0,
a x (1/a) = 1 & (1/a) x a = 1, shows 1/a is the
inverse of a for multiplication.
Hence proved…
NCERT MATHEMATICS
TEXTBOOK FOR CLASS XII…
PART I