Can you assist me with a few logic questions?

**doublecheckmyanswerspleas** · October 8th 2008, 10:08 AM

Okay, for my first question, I dont understand how the following statement is equal to the other:

the statement : (P-->Q)---> ~(~Q-->P)

is equal to: (P^Q) v (~Q ^~P)

How?

**Soroban** · October 8th 2008, 10:59 AM

Hello, doublecheckmyanswerspleas!

I suspect there's a typo . . .

I dont understand how the following statement is equal to the other:

The statement: . $(P \to \:{\color{red}\sim}\,Q) \to \;\sim(\sim\!Q \to P)$

is equal to: . $(P \wedge Q) \vee (\sim\!Q \:\wedge \sim\!P)$

$p \to q \;\equiv \;\sim p \vee q$ . . I call it ADI (alternate definition of implication).

$\begin{array}{ccccc} (P \to \:\sim Q) \; \to \; \sim(\sim\!Q \to P) & & \text{Given} \\ \\ (\sim\!P \:\vee \sim\!Q) \;\to \;\sim(Q \vee P) & & \text{ADI} \\ \\ (\sim\!P \:\vee \sim\!Q) \;\to \;(\sim\!Q \:\wedge \sim\!P) & & \text{DeMorgan} \\ \\ \sim(\sim\!P\:\vee \sim\!Q) \;\vee \;(\sim\!Q \:\wedge \sim\!P) & & \text{ADI} \\ \\ (P \wedge Q) \;\vee \;(\sim\!Q \:\wedge \sim\!P)& & \text{DeMorgan} \end{array}$

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