How many commutative binary operations can be defined on a set with 5 element ?

Verify whether following operation is commutative and associative:

 $a*b=a-b-ab$.  $\forall a,b \in \mathrm{\mathbb{Z}}$

Verify whether following operation is commutative and associative:

$a*b=a÷b$   $\forall a,b\in {\mathrm{\mathbb{Q}}}^{*}$

Verify whether following operation is commutative and associative:

$a*b=\frac{a}{b}$   $\forall a,b \in {\mathrm{ℝ}}^{*}$

Verify whether following operation is commutative and associative:

$a*b=\min \left(a,b\right)$   $\forall a,b \in \mathrm{ℝ}$

Verify whether following operation is commutative and associative:

$a*b=l.c.m\left(a,b\right)$    $\forall a,b\in {\mathrm{ℝ}}^{*}$

Given $S\ne \varphi ,$ consider the power set $P\left(S\right)$. Show that $P\left(S\right)$ is closed under formation of union. Show further that  $\left(P\left(S\right),\cup \right)$ is a commutative and associative system. Does there exist any identity in this system?

Let $S$$=\left\{1,2,3,4,5,\right\}$ and $*$ be a binary operation on $S$ defined by $a*b=\mathrm{g}.\mathrm{c}.\mathrm{d}.\left(a,b\right).$ Find the cayley table of $\left(S,*\right)$. 

Suppose $X=\varphi$$;$ Consider $P\left(X\right)$ and define $*$ on $P\left(X\right)$ by $A*B=A\cap B$$.$ Show that  $\left(P\left(X\right),*\right)$ is a commutative and associative system with identity.