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 Truth-tables 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 |
