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Bikram
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in Set Theory & Algebra
Aug 8, 2016

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** Commutative Operation:**

A binary operation * over a set N is said to be commutative, if for every pair of elements $a,b ∈ N$ ,$a*b=b*a$.

**Associative Operation:**

A binary operation * a on a set N is called associative if $a*(b*c)=(a*b)*c$ for all a,b,c ∈N.

**Addition(+)**

Addition operation on N (set of natural number) are commutative as well as associative.

Since for all a ,b ∈ N , a+b=b+a so addition on N is commutative.

Also $(a+b)+c=a+(b+c)$ for all a,b,c∈ N. so addition on N is associative .

**Multiplication**$\left ( \times \right )$

Multiplication operation on N (set of natural number) are also commutative as well as associative.

Since for all a ,b ∈ N , ab=ba so multiplication on N is commutative.

Also $(ab)c=a(bc)$ for all a,b,c∈ N. so multiplication on N is associative .

**Substraction**(-)

Substraction operation on N (set of natural number) is neither commutative nor associative.

Since $a-b = b -a$ and $(a-b)-c=a-(b-c)$ cannot be true for every pair of natural numbers a and b.

Eg.$5-4\neq 4-5$ and $(5-4)-3\neq 5-(4-3)$

**Division**($\div$)

Division operation on N (set of natural number) is also neither commutative nor associative.

Since $a \div b = b \div a$ and $(a \div b) \div c=a \div (b \div c)$ cannot be true for every pair of natural numbers a and b.

Eg:$3\div 4\neq 4\div 3$ and $(3\div 4)\div 1\neq 3\div (4\div 1)$.

So we can say substraction and division on N (set of natural number) are neither commutative nor associative.