Boolean Logic and Truth Tables

Question

I have been Googling around and haven't been able to find a solution. If anyone can link me or explain this, I'd appreciate it.

I have this expression:

¬aΛb | aΛ¬b. Λ is AND, ¬ is NOT.

The truth table is:

A B Expression
--------------
T T F
T F T
F T F
F F T

I am confused as to why they aren't all FALSE. For example, if I were to consider a and b as false: ¬a and ¬b gets precedence, so they become true. But ¬a (TRUE) Λ b (FALSE) is FALSE. And since Λ gets precedence, a (FALSE) Λ ¬b (TRUE) is again FALSE. So FALSE | FALSE = FALSE, right?

Likewise, for a|b|c|d|e, where | is OR. Why is it that when only d is FALSE, and the other are true:

T   T   T   F   T  

= FALSE

Answer 1

The calculator you're using uses | to mean NAND, not OR. You should use + for OR. Then the truth table comes out as expected. x NAND y is TRUE except when x AND y are true; and NAND has the same precedence as AND, so without parentheses the operators bind leftmost first. A fully parenthesized version of your formula is:

((((not a) and b) nand a) and (not b))

Generating a truth table based on this gives the observed result.