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A Binary Operation on a Set

ABSALG-11JGY2

A binary operation on a set $X$ can be thought of as function $f:X\times X\rightarrow X$.

From this viewpoint, how would the inverse of the element $x\in X$ be expressed?

Assume $e$ denotes the unique identity element.

A

There exists an element $y\in X$ such that $f(x,y)=f(y,x)=e$.

B

There exists elements $y,z\in X$ such that $f(x,y)=f(z,x)=e$.

C

There exists an element $y\in X$ such that $f(x,y)=e$.

D

There exists an element $y\in X$ such that $f(y,x)=e$.

E

None of the above.