Truth Tables for Validity: Using Equations to Determine Satisfaction

[XodoX]

I want to use truth tables to show that equations can be satisfied or not, or if they are valid.

not(X→(Y→X))

(X∧(notX→notY))→Y

I would say the first one is valid, because of the not in front of it, it's always true. I don't know about the second one. I don't know how to split them up best to use a truth table. I guess I can/should use:

X, notX, notY, Y, X∧(notX→notY), notX→notY and (X∧(notX→notY))→Y.

 

[Geofleur]

Here is one way to look at the problem: If you do a truth table for $\neg (A \rightarrow B)$, you will see that it is logically equivalent to the statement $A \& \neg B$. So your first statement is equivalent to $X \& \neg (Y \rightarrow X)$. But then that is equivalent to $X \& Y \& \neg X$. The basic idea is to use truth table identities to transform the complex statement into simpler ones.

 

[willem2]

You can make a truth table with a row for all combinations of y and x. As the columns you use x, y, (y->x), (x->(y->x)) and not(x->(y->x))

x y    (y->x)    x->(y-x)  not(x->(y-x))
0 0
0 1
1 0
1 1

If there are only ones in al column, the expression is always true, and if there are only zeros, the expression is never true for any x or y.