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Consider a binary operation * on the set $\{1,2,3,4,5\}$ given by the following multiplication table.
(i) Compute $(2 * 3) * 4$ and $2 *(3 * 4)$
(ii) Is * commutative?
(iii) Compute $\left(2^* 3\right)^*\left(4^{* 5}\right)$.
Hint: Use the following table.
(i) Compute $(2 * 3) * 4$ and $2 *(3 * 4)$
(ii) Is * commutative?
(iii) Compute $\left(2^* 3\right)^*\left(4^{* 5}\right)$.
Hint: Use the following table.
Solution:
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Verified Answer
(i) From the given table, we find
$2 * 3=1,1 * 4=1$
(a) $(2 * 3) * 4=1 * 4=1$
(b) $2 *(3 * 4)=2 * 1=1$
(ii) Let $a, b \in\{1,2,3,4,5\}$ From the given table, we find $a * a=a ~ a * b=b * a=1$ when $a$ or $b$ or are odd and $a \neq b$. $2 * 4=4 * 2=2$, when a and $b$ are even and $a \neq b$
Thus $a * b=b * c$
$\therefore$ Binary operation * given is commutative.
(iii) Binary operation * given is commutative
$$
(2 * 3) *(4 * 5)=1 * 1=1 \text {. }
$$
$2 * 3=1,1 * 4=1$
(a) $(2 * 3) * 4=1 * 4=1$
(b) $2 *(3 * 4)=2 * 1=1$
(ii) Let $a, b \in\{1,2,3,4,5\}$ From the given table, we find $a * a=a ~ a * b=b * a=1$ when $a$ or $b$ or are odd and $a \neq b$. $2 * 4=4 * 2=2$, when a and $b$ are even and $a \neq b$
Thus $a * b=b * c$
$\therefore$ Binary operation * given is commutative.
(iii) Binary operation * given is commutative
$$
(2 * 3) *(4 * 5)=1 * 1=1 \text {. }
$$
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