Verify the associative binary operation for the following function.  If it has associative binary operation find the identity element and inverse for every element with respect to that identity.

$\left(Z, *\right)$, where $a*b=a+b+1 \forall a, b\in Z$

Verify the associative binary operation for the following function.  If it has associative binary operation find the identity element and inverse for every element with respect to that identity.

$\left(Z, *\right)$, where $a*b=a+b-1 \forall a, b\in Z$

Verify the associative binary operation for the following function.  If it has associative binary operation find the identity element and inverse for every element with respect to that identity.

$\left(Z, *\right)$, where $a*b=a+b-ab \forall a, b\in Z$

Verify the associative binary operation for the following function.  If it has associative binary operation find the identity element and inverse for every element with respect to that identity.

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

Verify the associative binary operation for the following function.  If it has associative binary operation find the identity element and inverse for every element with respect to that identity.

$\left(\mathrm{ℝ}, *\right)$, where $a*b=a^2{b}^2 \forall a,b\in \mathrm{ℝ}.$

Verify the associative binary operation for the following function.  If it has associative binary operation find the identity element and inverse for every element with respect to that identity.

$\left(\mathrm{ℝ}, *\right)$, where $a*b=a+b+ab \forall a,b\in \mathrm{ℝ}.$

Verify the associative binary operation for the following function.  If it has associative binary operation find the identity element and inverse for every element with respect to that identity.

$\left({\mathrm{\mathbb{Q}}}^{+},*\right),$where $a*b$$=ab$$\forall a,b\in {\mathrm{\mathbb{Q}}}^{+}.$

Verify the associative binary operation for the following function.  If it has associative binary operation find the identity element and inverse for every element with respect to that identity.

$\left(\mathrm{\mathbb{Q}} \backslash \left\{0\right\} , *\right),$ where $a*b$$=ab$     $\forall a,b \in \mathrm{\mathbb{Q}} \backslash \left\{0\right\}.$

Let $X$ be a set such that $\left|X\right|=3.$How many different binary operations can be defined on $X$ ? How many of them are commutative binary operations?