If $p>q=p^2+q^3$ and $p<q=p^3-q^2$, then $(1>2)<3=$

1. First, find the value inside the parenthesis: $(1>2)$.
Here, $p=1$ and $q=2$. Since $1<2$, this expression actually checks the relationship or defined operation. Notice the definition uses the comparative symbols as operator symbols:

* When the operator written is '$>$ ': $p>q=p^2+q^3$
* When the operator written is '$<$ ': $p<q=p^3-q^2$

2. Therefore, for $(1>2)$, the operator is '$>$ ':

$$(1>2)=1^2+2^3=1+8=9$$

3. Now substitute this result back into the main expression:

$$(1>2)<3=9<3$$

4. For $9<3$, the operator used is '$<$ ':

$$9<3=9^3-3^2=729-9=720$$