Let * be a binary operation on N, the set of natural numbers, defined by a⋅b= ab for all a,b∈N˙ Is ′⋅′ associative or commutative on N?

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Let * be a binary operation on N, the set of natural numbers,

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Let * be a binary operation on N, the set of natural numbers, defined by $a \cdot b =$ $a^{b}$ for all $a, b \in N ˙$ Is $^{'} \cdot^{'}$ associative or commutative on $N ?$

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Theodore Discussed

Let * be a binary operation on N, the set of natural numbers, defined by

a \cdot b =

a^{b}

for all

a, b \in N ˙

^{'} \cdot^{'}

associative or commutative on

N ?

14 mins ago

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14 mins ago

Solution: Here,

a * b = a^{b}

for all

a, b \in N

.
Now,

a * b = a^{b} and b * a = b^{a}

∴ a * b \neq = b * a .

So, given operation is not associative.
Now,

(a * b) * c = (a^{b}) * c = (a^{b})^{c} = a^{b^{c}}

a * (b * c) = a * (b^{c}) = a^{b^{c}}

∴ (a * b) * c = a * (b * c) .

Therefore, given operation is commutative on

N

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Question Text	Let * be a binary operation on N, the set of natural numbers, defined by $a \cdot b =$ $a^{b}$ for all $a, b \in N ˙$ Is $^{'} \cdot^{'}$ associative or commutative on $N ?$
Answer Type	Text solution:1
Upvotes	150