Question

In a Group Z11 under multiplication operation the
inverse of 9 is​

Answer 1

$\textbf{Given:}$

$\mathsf{Z_{11}}$

$\textbf{To find:}$

$\textsf{Inverse of 9 under multiplication}$

$\textbf{Solution:}$

$\textbf{Inverse\;axiom:}$

$\boxed{\mathsf{a\,*\,a^{-1}=a^{-1}\,*\,a=e}}$

$\textsf{e-Identity element}$

$\mathsf{a^{-1}\;is\;the\;inverse\;of\;a}$

$\textsf{Consider,}$

$\mathsf{Z_{11}=\{0,1,2,3,4,5,6,7,8,9,10\}}$

$\mathsf{9\,._{11}\,1=9}$

$\mathsf{9\,._{11}\,2=7}$

$\mathsf{9\,._{11}\,3=5}$

$\mathsf{9\,._{11}\,4=3}$

$\boxed{\mathsf{9\,._{11}\,5=1}}$

$\implies\textsf{Inverse of 9 is 5}$

$\textbf{Find more:}$

Show that the set z of all integers form a group with respect to binary operation defined by a*b-a+b+1 is an abelian group

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Consider an algebraic system (Q,*), where Q is the set of all non-zero rational numbers and * is a binary operation defined by a * b =a + b - ab . Determine whether (Q,*) is a group.

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