Formal Verification Proofs

[h2p]

Hello, I am confused on much of the following...

Determine if the following formulas are satisfiable. P' is the negation of P.

(P→Q)→(Q'→P') <-- I read this as being P=true then Q=true, Q'=false then P'=false, so Q=true implies Q'=false???

(Q→R)→((P→Q)→(P→R)) <-- I read this as (Q→R)→(Q→R) because (P→Q)→(P→R) = (Q→R) right???

(P∧Q)→ (P∨Q) <-- AND versus OR ???

(Q→R)→((P∨Q)→(P∨R)) <--

I am probably going about this incorrectly... any help would be greatly appreciated.

TIA!

 

[pka]

Determine if the following formulas are satisfiable.

What does the term satisfiable(sic) mean?
Does it mean that for some assignment of truth values the statement is true?
Or does it mean that for for every assignment of truth values the statement is true?
In either case why not make a truth table and find out!

 

[soroban]

Hello, h2p!

What methods are you allowed to use?
\(\displaystyle \;\;\)(It sounds like you know <u>no</u> methods .)
Truth tables only? \(\displaystyle \;\)Algebraic proofs?

Determine if the following formulas are satisfiable.

\(\displaystyle (P\,\to\, Q)\,\to\,(Q'\,\to\,P')\)

I read this as being P=true then Q=true, Q'=false then P'=false,
so Q=true implies Q'=false? \(\displaystyle \;\)what?
This one is true: an implication implies its contrapositive.


\(\displaystyle (Q\to R)\,\to\,[(P\,\to\,Q)\,\to\,(P\,\to R)]\)

I read this as \(\displaystyle (Q\,\to\,R)\,\to\,[(Q\,\to\,R)\)
because \(\displaystyle (P\,\to\,Q)\,\to\,(P\,\to\,R)\:=\Q\,\to\,R)\), right?\(\displaystyle \;\) This too makes no sense.


\(\displaystyle (P\,\cap\,Q)\,\to\,(P\,\cup\,Q)\)
This happens to be true . . . but what kind of proof is expected?


Can't read this one: \(\displaystyle \,(Q\,to\,R)\,\to\,[(P\,?\,Q)]\,\to\,(P\,?\,R)]\)
Do you ever preview your stuff before you post it?

 

[pka]

(Q−>R)−>(Q−>R) because (P−>Q)−>(P−>R) = (Q−>R) right??? is not correct!

If we take the case \(\displaystyle P \equiv F\quad \& Q \equiv T\quad \& \quad R \equiv F\) then \(\displaystyle \left[ {(P \to Q) \to (P \to R)} \right] \equiv T\) !
BUT \(\displaystyle \left[ {Q \to R} \right] \equiv F\).

 

[h2p]

Determine if the following formulas are satisfiable.

What does the term satisfiable(sic) mean?
Does it mean that for some assignment of truth values the statement is true?
Or does it mean that for for every assignment of truth values the statement is true?
In either case why not make a truth table and find out!

The concept of satisfiable is mentioned above. Compared to the other answers given, the truth table seems to make the most sense from my perspective. I will ask for clarity on the "satisfiable" concept. Thanks!