• if p,q,r a re simple propositions with truth table values T,F and T then the truth value of (~P v q) ^  ~r ​​= P is ?????.....

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Truth} \mathrm{table} \mathrm{value} \mathrm{of} \mathrm{P}\hspace{0.17em}\mathrm{is} \mathrm{T} \\ \Rightarrow \mathrm{Truth} \mathrm{table} \mathrm{value} \mathrm{of} ~\mathrm{P} \mathrm{is} \mathrm{F}. \\ \mathrm{Truth} \mathrm{table} \mathrm{value} \mathrm{of} \mathrm{Q} \mathrm{is} \mathrm{F} \\ \mathrm{Truth} \mathrm{table} \mathrm{value} \mathrm{of} \mathrm{R}\hspace{0.17em}\mathrm{is} \mathrm{T} \\ \Rightarrow \mathrm{Truth} \mathrm{table} \mathrm{value} \mathrm{of} ~\mathrm{R}\hspace{0.17em}\mathrm{is} \mathrm{F}. \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.=\left(~\mathrm{P} \mathrm{v} \mathrm{Q}\right)\wedge ~\mathrm{R} \\ =\left(\mathrm{F} \mathrm{v} \mathrm{F}\right)\wedge \mathrm{F} \\ =\mathrm{F}\wedge \mathrm{F} \left[\mathrm{As} \mathrm{F} \mathrm{or} \mathrm{F} \mathrm{is} \mathrm{F}\right] \\ =\mathrm{F} \\ \mathrm{R}.\mathrm{H}.\mathrm{S}.=\mathrm{P}=\mathrm{T} \\ \Rightarrow \mathrm{L}.\mathrm{H}.\mathrm{S}.\ne \mathrm{R}.\mathrm{H}.\mathrm{S}. \\ \mathrm{Hence} \mathrm{truth} \mathrm{value} \mathrm{is} \mathrm{False} \end{gathered}$$

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