138 lines
3.8 KiB
Markdown
138 lines
3.8 KiB
Markdown
---
|
|
categories:
|
|
- Logic
|
|
- Computer Architecture
|
|
tags: [propositional-logic, algebra, nand-to-tetris]
|
|
---
|
|
|
|
# Boolean algebra
|
|
|
|
## Algebraic laws
|
|
|
|
Many of the laws that obtain in the mathematical realm of algebra also obtain for Boolean expressions.
|
|
|
|
### The Commutative Law
|
|
|
|
$$
|
|
x \land y = y \land x \\
|
|
$$
|
|
|
|
$$
|
|
|
|
x \lor y = y \lor x
|
|
$$
|
|
|
|
Compare the [Commutative Law](/Mathematics/Prealgebra/Whole_numbers.md#the-commutative-property) in the context of arithmetic.
|
|
|
|
### The Associative Law
|
|
|
|
$$
|
|
x \land (y \land z) = (x \land y) \land z
|
|
$$
|
|
|
|
$$
|
|
x \lor (y \lor z) = (x \lor y) \lor z
|
|
$$
|
|
|
|
Compare the [Associative Law](/Mathematics/Prealgebra/Whole_numbers.md#the-associative-property) in the context of arithmetic.
|
|
|
|
### The Distributive Law
|
|
|
|
$$
|
|
x \land (y \lor z) = (x \land y) \lor (x \land z)
|
|
$$
|
|
|
|
$$
|
|
x \lor (y \land z) = (x \lor y) \land (x \lor z)
|
|
$$
|
|
|
|
Compare how the [Distributive Law applies in the case of algebra based on arithmetic](/Mathematics/Prealgebra/Distributivity.md):
|
|
|
|
$$
|
|
a \cdot (b + c) = a \cdot b + a \cdot c
|
|
$$
|
|
|
|
### 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
|
|
|
|
Say we have the following expression:
|
|
|
|
$$
|
|
\lnot(\lnot(x) \land \lnot (x \lor y))
|
|
$$
|
|
|
|
We can employ DeMorgan's Laws to convert the second conjunct to a different form:
|
|
|
|
$$
|
|
\lnot (x \lor y) = \lnot x \land \lnot y
|
|
$$
|
|
|
|
So now we have:
|
|
|
|
$$
|
|
\lnot(\lnot(x) \land (\lnot x \land \lnot y ))
|
|
$$
|
|
|
|
As we have now have an expression of the form _P and (Q and R)_ we can apply the Distributive Law to simplify the brackets (_P and Q and R_):
|
|
|
|
$$
|
|
\lnot( \lnot(x) \land \lnot(x) \land \lnot(y))
|
|
$$
|
|
|
|
Notice that we are repeating ourselves in this reformulation. We have $\lnot(x) \land \lnot(x)$ but this is just the same $\lnot(x)$ by the principle of **idempotence**. So we can reduce to:
|
|
|
|
$$
|
|
\lnot(\lnot(x) \land \lnot(y))
|
|
$$
|
|
|
|
This gives our expression the form of the first DeMorgan Law ($\lnot (P \land Q)$), thus we can apply the law ($\lnot P \lor \lnot Q$) to get:
|
|
|
|
$$
|
|
\lnot(\lnot(x)) \lor \lnot(\lnot(y))
|
|
$$
|
|
|
|
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).
|