On the set Q of all ration numbers if a binary operation - is defined by a- b-a b 5- prove that - is associative on Q-

Let #160;a, #160;b, #160;c #8712;Q. #160;Then, #160;a #160;* #160;b #160;* #160;c=a #160;* #160;bc5 #160; #160; #160; #160; #1 ...

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Standard XII

Mathematics

Associative Law of Binary Operation

On the set Q ...

Question

On the set Q of all ration numbers if a binary operation * is defined by $a * b = \frac{a b}{5}$ , prove that * is associative on Q.

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Solution

$Let a, b, c \in Q . Then, a * (b * c) = a * (\frac{b c}{5}) = \frac{a (\frac{b c}{5})}{5} = \frac{a b c}{25} (a * b) * c = (\frac{a b}{5}) * c = \frac{(\frac{a b}{5}) c}{5} = \frac{a b c}{25} Therefore, a * (b * c) = (a * b) * c, \forall a, b, c \in Q .$

Thus, * is associative on Q.

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On the set Q of all ration numbers if a binary operation * is defined by a*b=ab5, prove that * is associative on Q.

Let a, b, c∈Q. Then, a * b * c=a * bc5 =abc55 =abc25a * b * c=ab5 * c =ab5c5 =abc25Therefore,a * b * c=a * b * c,∀ a, b, c∈Q. Thus, * is associative on Q.

On the set Q of all ration numbers if a binary operation * is defined by $a * b = \frac{a b}{5}$ , prove that * is associative on Q.

$Let a, b, c \in Q . Then, a * (b * c) = a * (\frac{b c}{5}) = \frac{a (\frac{b c}{5})}{5} = \frac{a b c}{25} (a * b) * c = (\frac{a b}{5}) * c = \frac{(\frac{a b}{5}) c}{5} = \frac{a b c}{25} Therefore, a * (b * c) = (a * b) * c, \forall a, b, c \in Q .$

Thus, * is associative on Q.