Study if a tree represents a tautological proposition

Question

I have difficulties to solve this excercise:

1. The following path of a given tree in customary infix notation: $$[(\neg(p\wedge q)\Rightarrow r)\wedge ((r\vee q)\Rightarrow s)]$$ represents a tautological proposition (do not make truth table).
2. It is not possible to go through the 1) tree in preorder.

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For the first one, I don't know why there is the data of how the tree is traversed (usual infix notation): is it because it is the classical (or only) way to reduce a complex proposition in classical logic? Because with another notation it is not valid (as it says in point 2., which I think is true).

However, it is clear that 1. is not a tautology, because if $q$ is true and $s$ is false, the expression is false. So this excercise is solved like this, or do we have to apply logical laws?: $$\begin{matrix} (\neg(p\wedge q)\Rightarrow r)\wedge ((r\vee q)\Rightarrow s)&\underbrace{\Leftrightarrow}_{\textrm{Conditional equiv.}\\\;\;\textrm{and involution}}\\ ((p\wedge q)\vee r)\wedge (\neg (r\vee q)\vee s)&\underbrace{\Leftrightarrow}_{\textrm{De Morgan}}\\ ((p\wedge q)\vee r)\wedge ((\neg r\wedge \neg q)\vee s)&\underbrace{\Leftrightarrow}_{\textrm{Distributive}}\\ (p\vee r)\wedge (q\vee r)\wedge (\neg r\vee s)\wedge (\neg q\vee s),& \end{matrix}$$ but from here I do not know how to continue.

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For the second I made the tree path:

Looking at the graph the preorder path would be $$\wedge\Rightarrow\neg\wedge pqr\Rightarrow\vee rqs,$$

and this in logic does not represent anything; it has no meaning, therefore we can't go through the tree in preorder.

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Any help would be appreciate!

Thanks!

Answer 1

Whence this nonsense? From a text on parsing statements?

The tree is traversed with infix notation because our logic is built using infix notation. Polish notation and reverse Polish notation use prefix notation and postfix notation respectively. VonNeuman notation, infix notation, requires infix trees. The same logic with Polish notation requires prefix trees and with reverse Polish notation requires postfix trees. p -> q, Cpq, pqC is VonNeuman, Polish, reverse Polish notation for 'p implies q', respectively.

Are natural languages infix? English seems to be.
Perhaps languages that place the verb last are postfix