# Predicates help

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• Sep 29th 2008, 08:20 AM
captainjapan
Predicates help
a)
Determine whether (p →q) ^ (¬p → q) ≡q.

b)
Find a proposition with three variables p, q, and r that is true when p and r are true and q is false, and false otherwise.
• Sep 29th 2008, 09:20 AM
Soroban
Hello, captainjapan!

Quote:

a) Determine whether: . $[(p \to q) \wedge (\sim p \to q)] \:\Longrightarrow \:q$

By propositional logic . . . you can supply the reasons:

$(p \to q) \wedge ( \sim p \to q)$

$(\sim p \vee q) \wedge (p \vee q)$

$(\sim p \wedge p) \vee q$

$f \vee q$

$q$

By truth table . . .

. . . $\begin{array}{|c|c|| ccccccccc|}
p & q & [(p & \to & q) & \wedge & (\sim p & \to & q)] & \to & q \\ \hline \hline
T & T & T & T & T & T & F & T & T & {\color{blue}T} & T \\
T & F & T & F & F & F & F & T & F & {\color{blue}T} & F \\
F & T & F & T & T & T & T & T & T & {\color{blue}T} & T \\
F & F & F & T & F & F & T & F & F & {\color{blue}T} & F \\ \hline
& & 1 & 2 & 1 & 3 & 1 & 2 & 1 & 4 & 1 \end{array}$