Let ‘*’ be defined on the set N. Which of the following are both commutative and associative?

The correct option is a*b=a+b.

To determine which operation defined on the set of natural numbers $N$ is both commutative and associative, let us analyze the properties of commutativity and associativity for the given operation $a*b=a+b$.

1. Commutative Property:
An operation $*$ is commutative if for any two natural numbers $a$ and $b$, we have:

$a*b=b*a$

For the operation defined by $a*b=a+b$:
The left-hand side is:

$a*b=a+b$

The right-hand side is:

$b*a=b+a$

Since addition of natural numbers is commutative, we know that $a+b=b+a$ for all $a,b\in N$. Therefore, we have:

$a*b=b*a$

Thus, the operation is commutative.

2. Associative Property:
An operation $*$ is associative if for any three natural numbers $a$, $b$, and $c$, we have:

$(a*b)*c=a*(b*c)$

Let us evaluate the left-hand side:

$(a*b)*c=(a+b)*c=(a+b)+c$

Now, let us evaluate the right-hand side:

$a*(b*c)=a*(b+c)=a+(b+c)$

Since addition of natural numbers is associative, we know that:

$(a+b)+c=a+(b+c)$

Therefore, we have:

$(a*b)*c=a*(b*c)$

Thus, the operation is associative.

Since the operation $a*b=a+b$ satisfies both properties, it is indeed both commutative and associative.