Name: Properties of Binary Operations
Uploaded: 2019-07-19
Duration: 12 min 41 s
Description: In this video, we will learn about binary operations. We will also understand properties followed by binary operations with the help of suitable examples.

Question

In case

*

is a binary operation in

a * b = m a - n b

on

Q^{+}

where

m

and

n \in N

, test whether it is

(i)

associative

(i i)

commutative. Test further if the identity element exists and the inverse element for any element of the respective set exists.

Accepted Answer

Given $' *'$ on $Q^{+}$ defined by $a * b = m a - n b$ where $m, n \in N$

First we have to check for commutativity of $*$ .

Let $a, b \in Q^{+}$

Then, $a * b = m a - n b$ and

$b * a = m b - n a$

Therefore $a * b \neq b * a$

Thus, $*$ is not commutative on $Z$ .

Now we have to check for associativity of $*$ .

Let $a, b, c \in Z$

Then, $(a * b) * c = (m a - n b) * c$

$= m (m a - n b) - n c = m^{2} a - m n b - n c$

$a * (b * c) = a * (2 b + 3 c)$

$= m a - n (m b - n c) = m a - m n b + n^{2} c$

Therefore $(a * b) * c \neq a * (b * c)$

Thus, $*$ is not associative on $Z$ .

Hence, identity element and inverse element does not exists.

In case $$ is a binary operation in $a b = m a - n b$ on $Q^{+}$ where $m$ and $n \in N$ , test whether it is $(i)$ associative $(i i)$ commutative. Test further if the identity element exists and the inverse element for any element of the respective set exists.

Important Questions on Relation and Function

Learn and Prepare for any exam you want !

In case * is a binary operation in a*b=ma-nb on Q+ where m and n∈N, test whether it is i associative ii commutative. Test further if the identity element exists and the inverse element for any element of the respective set exists.

Important Questions on Relation and Function

Learn and Prepare for any exam you want !

In case $$ is a binary operation in $a b = m a - n b$ on $Q^{+}$ where $m$ and $n \in N$ , test whether it is $(i)$ associative $(i i)$ commutative. Test further if the identity element exists and the inverse element for any element of the respective set exists.