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Write out the truth table for $ \lnot \bigl(P \Rightarrow \left(Q \Leftrightarrow R\right)\bigr) \Leftrightarrow \lnot R$.
Hint: $\land$ is AND. $\lor$ is OR. $\lnot$ is NOT. $\Rightarrow$ is IMPLIES. $\Leftrightarrow$ is IFF. $\oplus$ is XOR.
$C_1$$C_2$$C_3$$C_4$
$P$$Q$$R$$Q \Leftrightarrow R$$P \Rightarrow C_1$$ \lnot C_2$$ \lnot R$$C_3 \Leftrightarrow C_4$
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$\mathsf{T}$$\mathsf{F}$$\mathsf{T}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c2}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c3}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c4}}\hspace{35px}}~}$
$\mathsf{T}$$\mathsf{F}$$\mathsf{F}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c2}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c3}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c4}}\hspace{35px}}~}$
$\mathsf{F}$$\mathsf{T}$$\mathsf{T}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r4c0}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r4c1}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r4c2}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r4c3}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r4c4}}\hspace{35px}}~}$
$\mathsf{F}$$\mathsf{T}$$\mathsf{F}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r5c0}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r5c1}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r5c2}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r5c3}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r5c4}}\hspace{35px}}~}$
$\mathsf{F}$$\mathsf{F}$$\mathsf{T}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r6c0}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r6c1}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r6c2}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r6c3}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r6c4}}\hspace{35px}}~}$
$\mathsf{F}$$\mathsf{F}$$\mathsf{F}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r7c0}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r7c1}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r7c2}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r7c3}}\hspace{35px}}~}$$\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r7c4}}\hspace{35px}}~}$
$C_1$$C_2$$C_3$$C_4$
$P$$Q$$R$$Q \Leftrightarrow R$$P \Rightarrow C_1$$ \lnot C_2$$ \lnot R$$C_3 \Leftrightarrow C_4$
$\mathsf{T}$$\mathsf{T}$$\mathsf{T}$$[\mathsf{T}]$$[\mathsf{T}]$$[\mathsf{F}]$$[\mathsf{F}]$$[\mathsf{T}]$
$\mathsf{T}$$\mathsf{T}$$\mathsf{F}$$[\mathsf{F}]$$[\mathsf{F}]$$[\mathsf{T}]$$[\mathsf{T}]$$[\mathsf{T}]$
$\mathsf{T}$$\mathsf{F}$$\mathsf{T}$$[\mathsf{F}]$$[\mathsf{F}]$$[\mathsf{T}]$$[\mathsf{F}]$$[\mathsf{F}]$
$\mathsf{T}$$\mathsf{F}$$\mathsf{F}$$[\mathsf{T}]$$[\mathsf{T}]$$[\mathsf{F}]$$[\mathsf{T}]$$[\mathsf{F}]$
$\mathsf{F}$$\mathsf{T}$$\mathsf{T}$$[\mathsf{T}]$$[\mathsf{T}]$$[\mathsf{F}]$$[\mathsf{F}]$$[\mathsf{T}]$
$\mathsf{F}$$\mathsf{T}$$\mathsf{F}$$[\mathsf{F}]$$[\mathsf{T}]$$[\mathsf{F}]$$[\mathsf{T}]$$[\mathsf{F}]$
$\mathsf{F}$$\mathsf{F}$$\mathsf{T}$$[\mathsf{F}]$$[\mathsf{T}]$$[\mathsf{F}]$$[\mathsf{F}]$$[\mathsf{T}]$
$\mathsf{F}$$\mathsf{F}$$\mathsf{F}$$[\mathsf{T}]$$[\mathsf{T}]$$[\mathsf{F}]$$[\mathsf{T}]$$[\mathsf{F}]$