823.sagews

29. a.

[~B ^(A->B)] -> ~A

This is a Implication, ~A is already negative and on one side by itself of the connective.

This means we will need a FALSE-> FALSE for a Tautology.

import sage.logic.propcalc as propcalc
One = propcalc.formula("(A->B)");
One.truthtable();

A B value False False True False True True True False False True True True

This Truth Table confims the following values may be used within (A->B):

A      B      value
False  False  True   
False  True   True   
True   True   True
We already see we have TRUE -> TRUE = TRUE. Lets simplify and place A to represent this confirmation and continue on.

import sage.logic.propcalc as propcalc
Two = propcalc.formula("(~B & A)");
Two.truthtable();

B A value False False False False True True True False False True True False

We see there is only one set of values for these Statement letters.

B      A      value 
False  True   True

We see that this is already our values. So we have only one last TruthTable to confirm our Tautology. Lets change (~B & A) to simply ~A as we know FALSE & TRUE = FALSE.

import sage.logic.propcalc as propcalc
Three = propcalc.formula("~A -> ~A");
Three.truthtable();

A value False True True True

There is our confirmation FALSE -> FALSE = Tautology as it is TRUE always.