# Predicates help

• Sep 29th 2008, 07:20 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, 08:20 AM
Soroban
Hello, captainjapan!

Quote:

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

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

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

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

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

$\displaystyle f \vee q$

$\displaystyle q$

By truth table . . .

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