Determine the truth- value of the statement.

[Guest]

Given p is false, q is true and r is false, determine the truth- value of the statement.

( ~ q V r ) ? ( ~ p ? q)

[~ r ? (p V ~q) ] ? (p? r)

 

[DrSteve]

The first one is

\(\displaystyle (\neg T\vee F)\rightarrow (\neg F\wedge T)\)

\(\displaystyle (F\vee F)\rightarrow (T\wedge T)\)

\(\displaystyle F\rightarrow T\)

\(\displaystyle T\)

 

[DrSteve]

Here's the second one.

\(\displaystyle [\neg F \rightarrow (F\vee \neg T)]\leftrightarrow (F\rightarrow F)\)

\(\displaystyle [T\rightarrow (F\vee F)]\leftrightarrow T\)

\(\displaystyle (T\rightarrow F)\leftrightarrow T\)

\(\displaystyle F\leftrightarrow T\)

\(\displaystyle F\)

 

[soroban]

Hello, Magical_Star!

Given \(\displaystyle p\) is false, \(\displaystyle q\) is true, and \(\displaystyle r\) is false,
. . determine the truth value of the given statement.

. . \(\displaystyle (\sim q \vee r )\:\to \\sim p \wedge q)\)

This is just one row of a truth table . . .

. . \(\displaystyle \begin{array}{|c|c|c||c|c|c|c|c|c|c|} p & q & r & (\sim q & \vee & r) & \to & (\sim p & \wedge & q \\ \hline F&T&F & F&F&F&{\bf T} &T&T&T \\ \hline &&& _1 & _2 & _1 & _{\bf3} & _1 & _2 & _1 \\ \hline \end{array}\)

\(\displaystyle [\sim r \to (p \:\vee \sim q) ]\;\leftrightarrow\p \to r)\)

. . \(\displaystyle \begin{array}{|c|c|c|| c|c|c|c|c|c|c|c|c|} p & q & r & [\sim r & \to & (p & \vee & \sim q)] & \leftrightarrow & (p & \to & r) \\ \hline F&T&F & T&F&F&F&F& {\bf F} &F&T&F \\ \hline & & & _1 & _3 & _1 & _2 & _1 & _{\bf4} & _1 & _2 & _1 \\ \hline \end{array}\)