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ece124: add nand nor

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eggy 2023-01-19 10:08:19 -05:00
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@ -38,7 +38,7 @@ The **OR** operator returns true if and only if **at least one** argument is tru
$$A+B$$
<img src="https://upload.wikimedia.org/wikipedia/commons/1/16/OR_ANSI_Labelled.svg" align="middle" width=200>(Source: Wikimedia Commons)</img>
<img src="https://upload.wikimedia.org/wikipedia/commons/1/16/OR_ANSI_Labelled.svg" width=200>(Source: Wikimedia Commons)</img>
The **NOT** operator returns the opposite of its singular input.
@ -46,6 +46,29 @@ $$\overline A \text{ or } A'$$
<img src="https://upload.wikimedia.org/wikipedia/commons/6/60/NOT_ANSI_Labelled.svg" width=200>(Source: Wikimedia Commons)</img>
The **NAND** operator is equivalent to **NOT AND**.
$$\overline{A\cdot B}$$
<img src="https://upload.wikimedia.org/wikipedia/commons/e/e6/NAND_ANSI_Labelled.svg" width=200>(Source: Wikimedia Commons)</img>
The **NOR** operator is equivalent to **NOT OR**.
$$\overline{A+B}$$
<img src="https://upload.wikimedia.org/wikipedia/commons/c/c6/NOR_ANSI_Labelled.svg" width=200>(Source: Wikimedia Commons)</img>
### NAND/NOR completeness
NAND and NOR are **universal gates** — some combination of them can form any other logic gate. Constructions of other gates using only these gates are called **NAND-NAND realisations** or **NOR-NOR realisations**.
This is useful in SOP as if two ANDs feed into an OR, all can be turned into NANDs to achieve the same result.
!!! example
NOT can be expressed purely with NAND as $A$ NAND $A$:
<img src="https://upload.wikimedia.org/wikipedia/commons/3/3f/NOT_from_NAND.svg" width=150>(Source: Wikimedia Commons)</img>
### Postulates
In binary algebra, if $x,y,z\in\mathbb B$ such that $\mathbb B=\{0, 1\}$: