Explain an Empty Relation.

If $A=\left\{a,b,c,d\right\}$ and $B=\left\{x,y,z,w\right\}$, then state with reason if the following is a relation from $A$ to $B.$

$R_2=\left\{\left(x,a\right),\left(b,y\right),\left(z,c\right)\right\}$

If $A=\left\{a,b,c,d\right\}$ and $B=\left\{x,y,z,w\right\}$, then state with reason if the following is a relation from $A$ to $B.$

$R_2=\left\{\left(x,a\right),\left(b,y\right),\left(z,c\right)\right\}$

If $A=\left\{a,b,c,d\right\}$ and $B=\left\{x,y,z,w\right\}$, then state with reason if the following is a relation from $A$ to $B.$

$R_3=\left\{\left(a,y\right),\left(b,x\right),\left(c,w\right)\right\}$

If $A=\left\{a,b,c,d\right\}$ and $B=\left\{x,y,z,w\right\}$, then state with reason if the following is a relation from $A$ to $B.$

$R_4=\left\{\left(a,y\right),\left(z,b\right),\left(x,c\right),\left(c,w\right)\right\}$

Let $A=\left\{1,2\right\}$ and $B=\left\{3,4\right\}$ and let $R=\left\{\left(a,b\right):a\in A,b\in B, a>b\right\}.$ Show that $R$ is an empty relation from $A$ to $B.$

Let $A=(1, 3, 4)$ and $B (2, 5, 7)$. Find the number of relations from $A$ to $B$.