logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
0 votes

For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,a*b\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,a*b\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,a*b\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, a*b= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,a*b\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, a*b= \frac {a} {b+1} & \;\end{array}$

Can you answer this question?
 
 

5 Answers

0 votes
Toolbox:
  • 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $a,b \in x$
  • 2) A binary operation $*A \in A \to A$ is said to be associative. if $ (a \times b) * c=a* (b*c)$ for all $a,b,c \in A$
(ii)
on Q $*$ defined by $a*b=ab+1$
 
$a*b=ab+1$
 
$b*a=ba+1=ab+1$
 
$ a,b \in Q \qquad ab=ba$
 
Therefore $a*b=b*a$
 
operation * is commutative
 
we see that
 
$(1*2)*3=(1 \times 2 +1)*3$
 
$=3*3=3 \times 3=3 \times 3 +1=10$
 
$1*(2*3)=1*(2 \times 3+1)=1*7=1 \times +1=8$
 
$(1*2)*3 \neq 1*(2*3)$
 
operation * is not associative

 

 

answered Feb 26, 2013 by meena.p
 
0 votes
Toolbox:
  • 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $a,b \in x$
  • 2) A binary operation $*A \in A \to A$ is said to be associative. if $ (a \times b) * c=a* (b*c)$ for all $a,b,c \in A$
(iii)
on Q $*$ defined by $a*b=\frac{ab}{2}$
 
$a*b=\frac{ab}{2}$
 
$b*a=\frac{ba}{2}=\frac{ab}{2}$
 
for $ a,b \in Q \qquad ab=ba$
 
Therefore $a*b=b*a$
 
operation * is commutative
 
$a,b,c \in Q$
 
$(a*b)*c=(\frac{ab}{2})*c$
 
$=\large\frac{(\frac{bc}{2})}{2}=\frac{abc}{4}$
 
$a*(b*c)=a*\frac{bc}{2}$
 
$=\large\frac{a(\frac{bc}{2})}{2}=\frac{abc}{4}$
 
$(a*b)*c=a*(b*c)$
 
Hence $*$ operation is associative

 

 

answered Feb 26, 2013 by meena.p
 
0 votes
Toolbox:
  • 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $a,b \in x$
  • 2) A binary operation $*A \in A \to A$ is said to be associative. if $ (a \times b) * c=a* (b*c)$ for all $a,b,c \in A$
(iv)
on $z^+ *$ defined by $a*b=2^{ab}$
 
$a*b=2^{ab}$
 
$b*a=2^{ba}=2^{ab}$
 
for $ a,b \in Z^+; \qquad ab=ba$
 
Therefore $a*b=b*a$
 
operation * is commutative
 
$1,2,3 \in Z^+$
 
we see that $(1*2)*3=2^{1 \times 2} * 3$
 
$=4*3 =2^4*3=2^{12}$
 
$1*(2*3)=1*2^{2 \times 3}$
 
$=1*2^6=1*64$
 
$=2^{1 \times 64}=2^{64}$
 
Therefore $(1*2)*3 \neq 1*(2*3)$
 
Therefore $*$ opeation is not associative

 

 

answered Feb 26, 2013 by meena.p
 
0 votes
Toolbox:
  • 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $a,b \in x$
  • 2) A binary operation $*A \in A \to A$ is said to be associative. if $ (a \times b) * c=a* (b*c)$ for all $a,b,c \in A$
(v)
on $z^+ *$ defined by $a*b=a^b$
 
we see that
 
$1*2=1^2=1$
 
$ 2*1=2^1=2$
 
Therefore $1*2 \neq 2*1$
 
Hence operation * is not commutative
 
we also see that
 
$=4*3 =2^4*3=2^{12}$
 
$(2*3)*4=2^3*4$
 
$=8*4$
 
$=8^4$
 
$2*(3*4)=2*3^4$
 
$=2*81$
 
$=2^{81}$
 
$(2*3)*4 \neq 2*(3*4)$
 
Hence $*$ opeation is not associative

 

 

answered Feb 26, 2013 by meena.p
 
0 votes

Toolbox:

  • 1) A binary operation $*$ on set x is called commutative if $a*b=b*a$ for every $a,b \in x$
  • 2) A binary operation $*A \in A \to A$ is said to be associative. if $ (a \times b) * c=a* (b*c)$ for all $a,b,c \in A$
(vi)
on $R-\{-1\} *$ defined by $a*b=\frac{a}{b+1}$
 
we see that
 
$1*2=\frac{1}{2+1}=\frac{1}{3}$
 
$ 2*1=\frac{2}{1+1}=\frac{2}{2}=1$
 
$1*2 \neq 2*1 $
 
Hence operation * is not commutative
 
we see that
 
$(1*2)*3=\frac{1}{3}*3$
 
$=\frac{\frac{1}{3}}{3+1}=\frac{1}{12}$
 
$1*(2*3)=1*\frac{2}{3+1}$
 
$=1*\frac{1}{2}$
 
$=\large\frac{1}{\frac{1}{2}+1}=\frac{2}{3}$
 
$(1*2)*3 \neq 1*(2*3)$
 
Hence $*$ opeation is not associative

 

 

answered Feb 26, 2013 by meena.p
 

Related questions

Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...