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Logical Circuits

• Fundamentals
  • Propositional calculus
  • Semiconductors
  • MOSFETs
  • CMOS
• Circuits
  • Identity
  • NOT
  • AND
  • NAND
  • OR
  • XOR
  • XNOR
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XOR (eXclusive OR, "either .. or")

Introduction

The circuit of the "exclusive or" has two inputterminals, which are linked. The output is 1 ("true"), if there is an odd number of ones at the inputs. The meaning of the "either .. or" shows it well, "either" there is one inputterminal true "or" the other one. They have to be different for getting an 1 (true) as a result.

Example

claim: Either I or you win.
(The claim is false if both or nobody of them wins.)

If ...	the claim is ...
Nobody wins,	wrong
you win,	true
I win,	true
We both win,	false

claim (Z) = Either I(A) or you(B) win.

A	B	Z
loses	loses	false
loses	wins	true
wins	loses	true
wins	wins	wrong

Z = I XOR you (wins)
1: victory/ true
0: defeat/ wrong

Truth table

A XOR B = Z

A	B	Z
0	0	0
0	1	1
1	0	1
1	1	0

Functional equation

Functional symbol

" =1 " (if there is exactly one 1 (with 2 inputterminals), then true)

Exercises

Propositional calculus with words:

claim: Either it rains or the sun shines.

If the sun shines AND it's not raining, the claim is _________.
true
If the sun shines and it's raining _________.
wrong

From words to the logic function:

Either green or red.
green XOR red
Either you go or you don't go.
go XOR !go

Specials

It possible to have an XOR gate with more than two inputterminals the result is calculated in the same way like a gate with two inputterminal. The result is also true, if there is an odd number of inputterminals with an one. It's implemented with many XOR gates, where two inputterminals are connected with a xor and the result is (xor) connected with the next input. This process is repeated until every input is considered.

addition of binary numbers in the Von Neumann architecture

exlusive or	addition	result
0 XOR 0	0 + 0	0
0 XOR 1	0 + 1	1
1 XOR 0	1 + 0	1
1 XOR 1	1 + 1	(1)0

more information

encryption: One-Time-Pad
(plaintext → binary text XOR random key (same length as binary text)= secret message;
secret message XOR key = binary text → plaintext)
more information

It is also possible to build this gate just from NAND gates.

CMOS implementation