 stratsticks57jl

2022-07-16

Rewriting biconditional statements into conditional statement without arrows (discrete math)
I'm stuck on rewriting biconditional statements to conditional statements without using arrows. the original is as follows:
$\left(\left(P\to \left(\sim Q\right)\right)↔r\right)$ and what i (believe) the correct answer is:
$\left(\left(\sim PV\sim Q\right)↔r\right)$
$\left[\sim \left(\sim PV\sim Q\right)Vr\right]\Lambda \left[\left(\sim rV\left(\sim PV\sim Q\right)\right]$If anyone would like to correct and explain why i'm wrong or verify it's correct I would greatly appreciate it. Sandra Randall

Expert

Step 1
Statement $A⇒B$ can be reduced to $B\vee \phantom{\rule{thickmathspace}{0ex}}\overline{A}$
And $P\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}Q$ is equivalent to $\left(P⇒Q\right)\wedge \left(Q⇒P\right)$
Step 2
So, $\left(\left(P⇒\overline{Q}\right)\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}r\right)$ is equivalent to
- $\overline{Q}\phantom{\rule{thickmathspace}{0ex}}\vee \overline{P}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}r$
- $\left(r\vee \overline{\overline{Q}\phantom{\rule{thickmathspace}{0ex}}\vee \overline{P}}\right)\wedge \left(\left(\overline{Q}\phantom{\rule{thickmathspace}{0ex}}\vee \overline{P}\right)\vee \overline{r}\right)$