Question

Show that the propositions p1,p2,p3,p4, and p5 can be shown to be equivalent by proving that the conditional statements p1→p4, p3→p1, p4→p2, p2→p5, and p5→p3 are true.

Answer 1

$Suppose\,\,p_1=1.\\Then\\p_1\land \left( p_1\rightarrow p_4 \right) =p_1\land \left( \lnot p_1\lor p_4 \right) =p_1\land p_4=p_4\Rightarrow p_4=1\\p_4\land \left( p_4\rightarrow p_2 \right) =p_4\land \left( \lnot p_4\lor p_2 \right) =p_4\land p_2=p_2\Rightarrow p_2=1\\p_2\land \left( p_2\rightarrow p_5 \right) =p_2\land \left( \lnot p_2\lor p_5 \right) =p_2\land p_5=p_5\Rightarrow p_5=1\\p_5\land \left( p_5\rightarrow p_3 \right) =p_5\land \left( \lnot p_5\lor p_3 \right) =p_5\land p_3=p_3\Rightarrow p_3=1\\In\,\,this\,\,case\,\,p_1=p_2=p_3=p_4=p_5=1\\Suppose\,\,p_1=0.\\Then\\\lnot p_1\land \left( p_3\rightarrow p_1 \right) =\lnot p_1\land \left( \lnot p_3\lor p_1 \right) =\lnot p_1\land \lnot p_3=\lnot p_3\Rightarrow p_3=0\\\lnot p_3\land \left( p_5\rightarrow p_3 \right) =\lnot p_3\land \left( \lnot p_5\lor p_3 \right) =\lnot p_3\land \lnot p_5=\lnot p_5\Rightarrow p_5=0\\\lnot p_5\land \left( p_2\rightarrow p_5 \right) =\lnot p_5\land \left( \lnot p_2\lor p_5 \right) =\lnot p_5\land \lnot p_2=\lnot p_2\Rightarrow p_2=0\\\lnot p_2\land \left( p_4\rightarrow p_2 \right) =\lnot p_2\land \left( \lnot p_4\lor p_2 \right) =\lnot p_2\land \lnot p_4=\lnot p_4\Rightarrow p_4=0\\In\,\,this\,\,case\,\,p_1=p_2=p_3=p_4=p_5=0\\In\,\,both\,\,cases\,\,p_1=p_2=p_3=p_4=p_5, which\,\,means\,\,the\,\,propositions\,\,are\,\,equivalent.$