rewriting biconditional statements into conditional statement without arrows (discrete math)

i'm currently taking discrete math and my study guide gives the problems without the answers. (very inconvenient).. but anyways i'm stuck on rewriting biconditional statements to conditional statements without using arrows. the original is as follows:

((P→(~Q))<-->r)

& what i (believe) the correct answer is:

((~P V ~Q) <--> r)
[~(~P V ~Q) V r] Λ [(~r V (~P V ~ Q)]

If anyone would like to correct and explain why i'm wrong or verify it's correct I would greatly appreciate it. Thank You. :)

1 Answer

Statement $A \Rightarrow B$ can be reduced to $B \vee\;\bar{A}$

And $P \iff Q$ is equivalent to $(P \Rightarrow Q) \wedge (Q \Rightarrow P)$

So, $((P \Rightarrow \bar{Q}) \iff r )$ is equivalent to

• $\bar{Q}\;\vee \bar{P} \iff r $
• $(r \vee \overline{\bar{Q}\;\vee \bar{P}}) \wedge ((\bar{Q}\;\vee\bar{P}) \vee \bar{r})$

So I think your answer is correct.