Q. 14  Define a binary operation ∗ on the set \{0, 1, 2, 3, 4, 5\} as

a * b = \left\{\begin{matrix} a + b &if\;a+b < 6 \\ a+ b -6 & if\;a+b\geq6 \end{matrix}\right.

Show that zero is the identity for this operation and each element a\neq 0of the set
is invertible with 6 - a being the inverse of a.

Answers (1)

X =\{0, 1, 2, 3, 4, 5\} as 

 a * b = \left\{\begin{matrix} a + b &if\;a+b < 6 \\ a+ b -6 & if\;a+b\geq6 \end{matrix}\right.

An element  c \in X   is identity element for operation *,  if a*c=a=c*a \, \, \forall \, \, a \in X

For  a \in X,

               a *0 = a+0=a\, \, \, \, \, \, \, \, \, \, \, \, \left [ a \in X \Rightarrow a+0< 6 \right ]

              0 *a = 0+a=a\, \, \, \, \, \, \, \, \, \, \, \, \left [ a \in X \Rightarrow a+0< 6 \right ]

 \therefore \, \, \, \, \, \, \, \, \, a*0=a=0 *a   \forall a \in X

        Hence, 0 is identity element of operation *.

An element  a \in X  is invertible if there exists b \in X,

 such that   a*b=0=b*a      i.e. \left\{\begin{matrix} a + b =0=b+a &if\;a+b < 6 \\ a+ b -6=0=b+a-6 & if\;a+b\geq6\end{matrix}\right.

   means  a=-b     or    b=6-a

But since we have  X =\{0, 1, 2, 3, 4, 5\}  and a,b \in X. Then a\neq -b.

\therefore b=a-x   is inverse of a  for a \in X.

Hence, inverse of element  a \in X,a\neq 0   is  6-a   i.e. ,a^{-1} = 6-a

 

 

  

 

 

 

 

 

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