eolas/zk/Boolean_functions.md

tags
propositional-logic nand-to-tetris logic

Boolean functions

An example of a Boolean function:

f(x,y,z) = (x \land y) \lor (\lnot(x) \land z )

Here is a work through where f(1, 0, 1):

• The first disjunction : \lnot(x) \land z is false because x is 1 and z is 0
• The second disjunction: x \land y is false because x is 1 and y is 1
• The overall function returns false because the main connective is disjunction and both of its disjuncts are false

We can compute all possible outputs of the function by constructing a trkjuth table with each possible variable as the truth conditions and the output of the function as the truth value:

x	y	z	f(x,y,z) = (x \land y) \lor (\lnot(x) \land z )
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	0
1	1	0	1
1	1	1	1