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17
Logic/Propositional_logic/Boolean_algebra.md
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@ -91,12 +91,27 @@ $$
\lnot(\lnot(x)) \lor \lnot(\lnot(y))
$$
Of course now we have two double negatives. We can apply the double negation law to:
Of course now we have two double negatives. We can apply the double negation law to get:
$$
x \lor y
$$
### Truth table
Whenever we simplify an algebraic expression the value of the resulting expression should match that of the complex expression. We can demonstrate this with a truth table:
| $x$ | $y$ | $\lnot(\lnot(x) \land \lnot (x \lor y))$ | $x \lor y$ |
| --- | --- | ---------------------------------------- | ---------- |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 |
### Significance for computer architecture
The fact that we can take a complex Boolean function and reduce it to a simpler formulation has great significance for the development of computer architectures, specifically [logic gates](/Electronics_and_Hardware/Digital_circuits/Logic_gates.md). It would be rather resource intensive and inefficient to create a gate that is representative of the complex function. Whereas the simplified version only requires a single [OR gate](/Electronics_and_Hardware/Digital_circuits/Logic_gates.md#or-gate)
// TO DO:
- Use truth tables to show equivalence