Logic boolean algebra problem

Question

so I have to prove that these equations :

are equivalent?

Answer 1

This is a good example to get to know all boolean operators and to practice operator precedence.

I translated both expressions into boolean C# expressions:

u = !!(x || !y) || !(!x && z) || (!(!x && (y == z)) || (!(x && y) != z));
v = !((!x || y) && !(x || (y && !z))) || !(y && z);

The translation is not unique. It assumes certain operator priorities.

To do the translation, the following operators have to be taken into account:

When I run these expressions in a home-grown program, I get the following truth table:

This shows that both expressions are equivalent.

May be, that my assumptions are not true and the result should be a tautology for both expressions (all rows in the truth table have $1$ as result). This could lead to an equivalence proof: Show that both expressions are in fact tautologies.