Consider a binary operation $*$ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table. Compute $(2 \,^* \,3) \,^* \,4$ and $2 \,^* \,(3 \,^* \,4)$.
$^*$ $1$ $2$ $3$ $4$ $5$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $1$ $2$ $2$ $2$ $2$ $3$ $1$ $2$ $3$ $3$ $3$ $4$ $1$ $2$ $3$ $4$ $4$ $5$ $1$ $2$ $3$ $4$ $5$
| $^*$ | $1$ | $2$ | $3$ | $4$ | $5$ |
| $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $2$ | $1$ | $2$ | $2$ | $2$ | $2$ |
| $3$ | $1$ | $2$ | $3$ | $3$ | $3$ |
| $4$ | $1$ | $2$ | $3$ | $4$ | $4$ |
| $5$ | $1$ | $2$ | $3$ | $4$ | $5$ |
(N/A) To compute $(2 \,^* \,3) \,^* \,4$:
From the table,$2 \,^* \,3 = 2$.
Then,$(2 \,^* \,3) \,^* \,4 = 2 \,^* \,4 = 2$.
To compute $2 \,^* \,(3 \,^* \,4)$:
From the table,$3 \,^* \,4 = 3$.
Then,$2 \,^* \,(3 \,^* \,4) = 2 \,^* \,3 = 2$.
From the table,$2 \,^* \,3 = 2$.
Then,$(2 \,^* \,3) \,^* \,4 = 2 \,^* \,4 = 2$.
To compute $2 \,^* \,(3 \,^* \,4)$:
From the table,$3 \,^* \,4 = 3$.
Then,$2 \,^* \,(3 \,^* \,4) = 2 \,^* \,3 = 2$.
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