Let $*^{\prime}$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a *^{\prime} b = \text{H.C.F. of } a \text{ and } b$. Is the operation $*^{\prime}$ same as the operation $*$ defined in Exercise $4$ above? Justify your answer.
(B) The binary operation $*^{\prime}$ on the set $\{1, 2, 3, 4, 5\}$ is defined as $a *^{\prime} b = \text{H.C.F. of } a \text{ and } b$.
The operation table for the operation $*^{\prime}$ is given below:
In Exercise $4$,the operation $*$ is defined as $a * b = \min\{a, b\}$.
Comparing the tables,we observe that the operation tables for the operations $*$ and $*^{\prime}$ are different.
Thus,the operation $*^{\prime}$ is not the same as the operation $*$.
The operation table for the operation $*^{\prime}$ is given below:
| $*^{\prime}$ | $1$ | $2$ | $3$ | $4$ | $5$ |
| $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $2$ | $1$ | $2$ | $1$ | $2$ | $1$ |
| $3$ | $1$ | $1$ | $3$ | $1$ | $1$ |
| $4$ | $1$ | $2$ | $1$ | $4$ | $1$ |
| $5$ | $1$ | $1$ | $1$ | $1$ | $5$ |
In Exercise $4$,the operation $*$ is defined as $a * b = \min\{a, b\}$.
Comparing the tables,we observe that the operation tables for the operations $*$ and $*^{\prime}$ are different.
Thus,the operation $*^{\prime}$ is not the same as the operation $*$.
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