Answer
p $\rightarrow$ t $\qquad$ { by premise (d) }
$\sim$t $\qquad$ { by premise (c) }
$\therefore$ $\sim$p $\qquad$ { by modus tollens } $\longrightarrow$ Conclusion (1)
$\sim$p $\qquad$ { by Conclusion (1) }
$\therefore$ $\sim$p $\lor$ q $\qquad$ { by generalization rule } $\longrightarrow$ Conclusion (2)
$\sim$p $\lor$ q $\rightarrow$ r $\qquad$ {by premise (a) }
$\sim$p $\lor$ q $\qquad$ { by conclusion (2) }
$\therefore$ r $\qquad$ { by modus ponens } $\longrightarrow$ Conclusion (3)
$\sim$p $\qquad$ { by Conclusion (1) }
r $\qquad$ { by Conclusion (3) }
$\therefore$ $\sim$p $\land$ r $\qquad$ {by conjunction rule} $\longrightarrow$ Conclusion (4)
$\sim$p $\land$ r $\rightarrow$ $\sim$s $\qquad$ {by premise (e) }
$\sim$p $\land$ r $\qquad$ { by Conclusion (4) }
$\therefore$ $\sim$s $\qquad$ { by modus ponens } $\longrightarrow$ Conclusion (5)
s $\lor$ $\sim$ q $\qquad$ { by premise (b) }
$\sim$ s $\qquad$ { by Conclusion (5) }
$\therefore$ $\sim$q $\qquad$ { by elimination rule}
Work Step by Step
Steps have been shown above
