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Home » A binary operation on the set is defined as     Show that 0 is the identity for this operation and each element a has an inverse To find: identity and inverse element
A binary operation * on the set (0,1,2,3,4,5) is defined as

    \[a * b=\left\{\begin{array}{ll} a+b ; & \text { if } a+b<6 \\ a+b-6 ; & \text { if } a+b \geq 6 \end{array}\right.\]

Show that 0 is the identity for this operation and each element a has an inverse (6-\mathrm{a}) To find: identity and inverse element
Binary Operations | CBSE | Class 12 | Exercise 3A | Maths | RS Aggarwal
A binary operation * on the set (0,1,2,3,4,5) is defined as

    \[a * b=\left\{\begin{array}{ll} a+b ; & \text { if } a+b<6 \\ a+b-6 ; & \text { if } a+b \geq 6 \end{array}\right.\]

Show that 0 is the identity for this operation and each element a has an inverse (6-\mathrm{a}) To find: identity and inverse element

For a binary operation if a*e = a, then e s called the right identity

If \mathrm{e}^{*} \mathrm{a}=\mathrm{a} then \mathrm{e} is called the left identity

For the given binary operation,

\begin{array}{l} e^{*} b=b \\ \Rightarrow e+b=b \\ \Rightarrow e=0 \text { whicl } \\ b^{*} e=b \\ \Rightarrow b+e=b \end{array}

\Rightarrow \mathrm{e}=0 which is less than 6

b^{*} e=b

\Rightarrow b+e=b

\Rightarrow \mathrm{e}=0 which is less than 6

For the 2^{\text {nd }} condition,

\begin{array}{l} e^{*} b=b \\ \Rightarrow e+b-6=b \\ \Rightarrow e=6 \end{array}

But e=6 does not belong to the given set (0,1,2,3,4,5)

So the identity element is 0

An element c is said to be the inverse of a, if a^{*} c= e where e is the identity element

\begin{array}{l} a^{*} c=e \\ \Rightarrow a+c=e \\ \Rightarrow a+c=0 \\ \Rightarrow c=-a \end{array}

a belongs to (0,1,2,3,4,5),

– a belongs to (0,-1,-2,-3,-4,-5)

So c belongs to (0,-1,-2,-3,-4,-5),

So c=-a is not the inverse for all elements a

Putting in the 2^{\text {nd }} condition

\begin{array}{l} a^{*} c=e \\ \Rightarrow a+c-6=0 \\ \Rightarrow c=6-a \\ 0 \leq a<6 \\ \Rightarrow-6 \leq-a<0 \Rightarrow 0 \leq 6-a<60 \leq c<5 \end{array}

=> c belongs to the given set

Hence the inverse of the element a is (6-a)

i

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