Prove that all regions of the diagram are non$-$empty sets by providing example histories H$1-$H$12$

answer should be similar to the below

T$1=$ w$1[$x$]$ w$1[$y$]$ w$1[$z$]$ c$1$

T$2=$ r$2[$u$]$ w$2[$x$]$ r$2[$y$]$ w$2[$y$]$ c$2$

And also consider the following four histories over transactions T$1$ and T$2$:

H$7=$ w$1[$x$]$ w$1[$y$]$ r$2[$u$]$ w$2[$x$]$ r$2[$y$]$ w$2[$y$]$ c$2$ w$1[$z$]$ c$1$

H$8=$ w$1[$x$]$ w$1[$y$]$ r$2[$u$]$ w$2[$x$]$ r$2[$y$]$ w$2[$y$]$ w$1[$z$]$ c$1$ c$2$

H$9=$ w$1[$x$]$ w$1[$y$]$ r$2[$u$]$ w$2[$x$]$ w$1[$z$]$ c$1$ r$2[$y$]$ w$2[$y$]$ c$2$

H$10=$ w$1[$x$]$ w$1[$y$]$ r$2[$u$]$ w$1[$z$]$ c$1$ w$2[$x$]$ r$2[$y$]$ w$2[$y$]$ c$2$

H$7$ is not Recoverable $($RC$).$

H$8$ is Recoverable $($RC$)$ but not Avoids Cascading Aborts $($ACA$).$

H$9$ is Avoids Cascading Aborts $($ACA$)$ but not Strict $($ST$).$

H$10$ is Strict $($ST$).$