Solution to When considering the set of all the natural numbers (ℕ), show whether the mathematical operations … - Sikademy
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Archangel Macsika

When considering the set of all the natural numbers (ℕ), show whether the mathematical operations of addition, subtraction, multiplication and division are: (a) Associative (b) Commutative.

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(a) Since a+(b+c)=a+b+c=(a+b)+c for any a,b,c\in\N, we conclude that the operation of addition is associative on the set \N of natural numbers.

Since a\cdot(b\cdot c)=a\cdot b\cdot c=(a\cdot b)\cdot c for any a,b,c\in\N, we conclude that the operation of multiplication is associative on the set \N of natural numbers.

Since 4-(2-1)=3\ne 1=(4-2)-1, we conclude that the operation of subtraction is not associative on the set \N of natural numbers.

Since 8:(4:2)=4\ne 1=(8:4):2, we conclude that the operation of division is not associative on the set \N of natural numbers.


(b) Since a+b=b+a for any a,b\in\N, we conclude that the operation of addition is commutative on the set \N of natural numbers.

Since a\cdot b=b\cdot a for any a,b\in\N, we conclude that the operation of multiplication is commutative on the set \N of natural numbers.

Since 2-1=1\ne -1=1-2, we conclude that the operation of subtraction is not commutative on the set \N of natural numbers.

Since 8:4=2\ne 0.5=4:8, we conclude that the operation of division is not commutative on the set \N of natural numbers.

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Question ID: mtid-5-stid-8-sqid-745-qpid-630