A Tutorial On The Boolean Logic Gates and Their Quick Understanding!
by R. Steve Walz rstevew@armory.com
The Gates: BUFFER, INVERTER, AND, NAND, OR, NOR, XOR, XNOR , that's ALL!
Call them XORs and XNORs, it's more correct and consistent. The X means
that the state where all Inputs are 1 is *eXcluded* from being like
that same line of the OR and NOR, respectively, that's all!! You will
note that the A=1 B=1 lines of the XOR and XNOR are the ONLY lines
which are opposite the OR and NOR, both respectively!! ALSO, note that
all gates with an "N" in front of the primary proposition, BUFFER, AND,
OR XOR, have inverted entries for their Outputs from their primaries!
Okay, THE BOOLEAN TRUTH TABLE for ALL GATES!!
( na = Not Applicable: The BUFFER and INVERTER have only 1 INPUT )
+--------+----------------------------------------------------+
| INPUTS | OUTPUTS |
+--------+------------------+-----------+---------+-----------+
| B A | BUFFER INVERTER | AND NAND | OR NOR | XOR XNOR |
+--------+------------------+-----------+---------+-----------+
| na 0 | 0 1 | na | na | na |
| na 1 | 1 0 | na | na | na |
+--------+------------------+-----------+---------+-----------+
| 0 0 | na | 0 1 | 0 1 = 0 1 |
| 0 1 | na | 0 1 | 1 0 = 1 0 |
| 1 0 | na | 0 1 | 1 0 = 1 0 |
| 1 1 | na | 1 0 | 1 0 =#= 0 1 | <--"X"
+--------+------------------+-----------+---------+-----------+
"Driver" |\ "NOT Gate" |\
BUFFER A--| >-- INVERTER A--| >O-- <--Note: "BALL"="N"=INVERTER
___ |/ ___ ___ |/ ___ ___ ___
A--| \ A--| \ A--\ \ A--\ \ A--\\ \ A--\\ \
| AND )-- |NAND )O-- ) OR )-- )NOR )O-- ))XOR )-- ))XNOR)O--
B--|___ / B--|___ / B--/____/ B--/___ / B--//___ / B--//___ /
Note: A "BALL" on Outputs Shows NOT+AND=NAND , NOT+OR=NOR , NOT+XOR=XNOR
As for writing Boolean propositions in Boolean Algebra, actually, once
you learn the Gates and DeMorgan's Laws, then you can do most problems
mostly by inspection!! If we use "-" for an inversion or "NOT", and "*"
for AND, and "+" for OR, and "(+)" for XOR, then we can write these like
a simple equational algebra starting with the "Output" on one side and
the Gate Array that produces it on the other using the equals sign!
A AND B : A*B , A OR B : A+B , A XOR B : A(+)B , NOT(A AND B) : -(A*B)
and et cetera.
So XOR means, precisely, Either JUST One Input OR ANY Other Input = 1,
But NOT Both(or All). This lets us write:
XOR: A(+)B = (-A*B)+(A*(-B)) or XNOR: -(A(+)B) = -[(-A*B)+(A*(-B))]
Can you see why XOR is a difference detector and XNOR is a comparator??
Note that Gates CAN have MORE than 2 Inputs, and the same principles
apply. Namely, that for an AND or NAND, the only Inputs which result
in a 1 Output or 0 Output, respectively for those two, are the case in
Which the ALL(EVERY) Input = 1 is TRUE. The OR or NOR is like its simple
rule as well, that Either One Input must = 1 OR the OTHER(ANY) Input = 1
for its Output to = 1 . Thus there are many-Inputted forms of All Gates!
To "N" the Output, merely put the Input of an INVERTER on the Gate Output
and its Output IS the "N" version of BUFFER, AND, OR, and XOR Gates.
Thus, if you don't HAVE an XOR Gate, then you can MAKE one of ANDs, an OR,
and INVERTERS from the above recipe, as follows:
|\ ___
[A]-+-----| >O--[-A]--| \
| |/ | )--[(-A)*B]--+
| +--------[B]---|___ / | ___
| | +--\ \
| | ) )---[((-A)*B)+(A*(-B))]
| | ___ +--/___ / = A(+)B !!!
+--]--------[A]---| \ |
| |\ | )--[A*(-B)]--+
[B]----+--| >O--[-B]--|___ /
|/
So you see, those equations actually MEAN these Gates doing the work, and
the Output gets its meaning from the inner propositions OUTWARD!! There
are a list of AXIOMS for Boolean manipulations of equations in any Logic
book. They are very like their plain algebraic cousins!! And DeMorgan's
Laws look difficult as equations, and make perfect sense when you do them
as Gates. Here's the secret to remembering DeMorgan's Laws:
"Any AND or OR Gate-core can be seen to be its opposite core with all its
Inputs and Outputs which ARE INVERTED becoming UN-INVERTED, and all Inputs
or Outputs which are NOT INVERTED *BECOMING* INVERTED!!"
Examples:
+---------+---------INVERSIONS------------+
| | |
|\ v ___ | | ____
A---| >O--| \ v A----------\ \
|/ |NAND )O--- <== EQUALS ==> |\ v ) OR )----
B----------|___ / B---| >O--/____/
|/
+--+----------+---INVERSIONS
| | |
|\ v | ___ | ___
A---| >O------| \ v A----\ \
|/ __| |NAND )O--- <== EQUALS ==> ) OR )----
| +--|___ / B----/___ /
|\ v | ^^^^^ ^^^^^
B---| >O---+ NO INVERSIONS
|/
+------------------INVERSIONS-------------+---------+
| | |
|\ v ___ | ___ |
A---| >O--| \ A----------\ \ v
|/ | AND )---- <== EQUALS ==> |\ v )NOR )O---
B----------|___ / B---| >O--/___ /
|/
INVERSIONS-------+--+-----------+
| | |
NO INVERSIONS |\ v | |
vvvv vvv ----| >O----+ |
___ |/ __| | ___ |
----| \ | +--\ \ v
| AND )---- <== EQUALS ==> |\ v )NOR )O---
----|___ / ----| >O-------/___ /
|/
A quicker way to draw this stuff is to JUST USE BALLS for INVERSIONS!:
DeMorgan's Laws in Gate form:
( <==> = "Is Equivalent To", and in both directions! )
___ ___
A---| \ A--O\ \
| AND )--- <==> ) OR )O-- <-- A true AND Gate at left!
B---|___ / B--O/___ / (Yes, it looks like an NOR, but just
___ ___ think of "BALLs" as discrete "NOTs"!)
A--O| \ A---\ \
| AND )--- <==> ) OR )O--
B---|___ / B--O/___ /
___ ___
A---| \ A--O\ \
| AND )--- <==> ) OR )O--
B--O|___ / B---/___ /
___ ___
A--O| \ A---\ \
| AND )--- <==> ) OR )O-- <-- A true NOR Gate!
B--O|___ / B---/___ /
___ ___
A---| \ A--O\ \
| AND )O-- <==> ) OR )--- <-- A true NAND Gate at left!
B---|___ / B--O/___ /
___ ___
A--O| \ A---\ \
| AND )O-- <==> ) OR )---
B---|___ / B--O/___ /
___ ___
A---| \ A--O\ \
| AND )O-- <==> ) OR )---
B--O|___ / B---/___ /
___ ___
A--O| \ A---\ \
| AND )O-- <==> ) OR )--- <-- A true OR Gate
B--O|___ / B---/___ /
In OTHER words, swap OR core for AND, and put "BALLs" where there weren't
and take them away where they were, and you have converted an OR to AND
core proposition, or the reverse. These above are EQUALITIES, which have
the SAME TRUTH TABLE as the Gate Array!! For further fun, use DeMorgan and
"slide" the "BALLs" along a wire to where two meet and cancel and see what
results, or introduce two at opposite ends of a wire and see what DeMorgan
Law conversions can be accomplished to use up hex gate and quad gate chips
most efficiently!! Look at our first XOR made of an OR, 2 ANDs, and 2 NOTs
above and see if you can make it ONLY with either NAND or NOR gates. These
are the only two Gates which each have the property that ALL other Gates
can be made from them alone, all NANDs or all NORs, even any computer
architecture, functionally speaking!!
And even though DeMorgan's Laws don't cover the XOR and XNOR Gates:
There ARE even SOME interesting DeMorgan-esque features like this to the
XOR and XNOR Gates, but I'll leave that exploration to you out there with
idle time and a lot of napkins at some restaurant tonight!!
I'm getting tired of drawing!! ;-)
Take good care,
-Steve
--
Copyright (C) 1997 by Richard Steven Walz - Free Non-Commercial Use
All Commercial Rights Reserved by Author
--
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