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37
Logic/Propositional_logic/Boolean_algebra.md
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@ -45,13 +45,37 @@ $$
x \lor (y \land z) = (x \lor y) \land (x \lor z) x \lor (y \land z) = (x \lor y) \land (x \lor z)
$$ $$
Compare for instance how this applies in the case of [multiplication](/Mathematics/Prealgebra/Distributivity.md): Compare how the [Distributive Law applies in the case of algebra based on arithmetic](/Mathematics/Prealgebra/Distributivity.md):
$$ $$
a \cdot (b + c) = a \cdot b + a \cdot c a \cdot (b + c) = a \cdot b + a \cdot c
$$ $$
In addition we have [DeMorgan's Laws](/Logic/Laws_and_theorems.md/DeMorgan's_Laws.md) which express the relationship that obtains between the negations of conjunctive and disjunctive expressions ### Double Negation Elimination
$$
\lnot \lnot x = x
$$
### Idempotent Law
$$
x \land x = x
$$
> Combining a quantity with itself either by logical addition or logical multiplication will result in a logical sum or product that is the equivalent of the quantity
### DeMorgan's Laws
In addition we have [DeMorgan's Laws](/Logic/Laws_and_theorems.md/DeMorgan's_Laws.md) which express the relationship that obtains between the negations of conjunctive and disjunctive expressions:
$$
\lnot(x \land y) = \lnot x \lor \lnot y
$$
$$
\lnot (x \lor y) = \lnot x \land \lnot y
$$
## Applying the laws to simplify complex Boolean expressions ## Applying the laws to simplify complex Boolean expressions
@ -110,11 +134,4 @@ Whenever we simplify an algebraic expression the value of the resulting expressi
### Significance for computer architecture ### 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) 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
- Explicitly add implicit laws
- Link to deductive rules
- Link to digital circuits and NANDs as universal gates