Determine which of the following binary operations on the set $N$ are associative and which are commutative: $a \ast b = 1$ for all $a, b \in N$.
(A) $1$. Commutativity: $A$ binary operation $\ast$ on a set $N$ is commutative if $a \ast b = b \ast a$ for all $a, b \in N$. Here,$a \ast b = 1$ and $b \ast a = 1$. Since $1 = 1$,the operation is commutative.
$2$. Associativity: $A$ binary operation $\ast$ on a set $N$ is associative if $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in N$. Here,$(a \ast b) \ast c = 1 \ast c = 1$. Also,$a \ast (b \ast c) = a \ast 1 = 1$. Since $1 = 1$,the operation is associative.
Therefore,the operation is both associative and commutative.
$2$. Associativity: $A$ binary operation $\ast$ on a set $N$ is associative if $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in N$. Here,$(a \ast b) \ast c = 1 \ast c = 1$. Also,$a \ast (b \ast c) = a \ast 1 = 1$. Since $1 = 1$,the operation is associative.
Therefore,the operation is both associative and commutative.
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