For any three propositions p, q and r, the proposition $(\mathrm{p}\wedge \mathrm{q})\wedge (\mathrm{q}\wedge \mathrm{r})$ is true when

$(\mathrm{p}\wedge \mathrm{q})\wedge (\mathrm{q}\wedge \mathrm{r})$  If $\mathrm{p}\wedge \mathrm{q}\text{ is true, }\mathrm{q}\wedge \mathrm{r}\text{ is true}$

$\mathrm{p}\wedge \mathrm{q}\text{ is true, if both p and q are true statements}$

$\mathrm{q}\wedge \mathrm{r}\text{ is true if both q and r are true statements}$

  $\therefore \mathrm{all} \mathrm{the} \mathrm{three} \mathrm{statements} \mathrm{p},\mathrm{q},\mathrm{r} \mathrm{are} \mathrm{true}$