eolas/Hardware/Logic_Gates/Logic_gates.md

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Logic Electronics Hardware logic-gates

Logic gates

Logic gates are the basic building blocks of digital computing. A logic gate is an electrical circuit that has one or more than one input and only one output. The input controls the output and the logic is isomorphic with Boolean connectives defined in terms of truth-tables.

Truth tables

Truth-tables present the conditions under which logical propositions are true or false. To take the AND operator: AND evaluates to true if both of its constituent expressions are true, and false in any other circumstances (e.g. if one proposition is true and the other false (or vice versa) and if both propositions are false ).

This is most clearly expressed in the following truth table:

Truth table for AND

p    q   p & q
_    _   _____
t    t     t
t    f     f
f    t     f
f    f     f   

The negation operator (¬ or ~ ) switches the value of a proposition from true to false. When we put ~ before true it becomes false and when we put ~ before false it becomes true. We will see shortly that this corresponds to a basic on/off switch.

Truth table fo NOT

p    ~ p
_    __
t    f
f    t

AND gate

Just as we can create NOT logic from a NAND gate, without the AND conditions, we can create a circuit that exemplifies the truth conditions of AND without including those of NOT.

When we attach two NAND gates in sequence connected to two switches as input this creates the following binary conditions:

A    B   Output
_    _   _____

0    0     0       (1)
1    0     0       (2)
0    1     0       (3)
1    1     1       (4)

Which is identical to the truth table for AND :

p    q   p & q
_    _   _____

t    t     t      (1)
t    f     f      (2)
f    t     f      (3)
f    f     f      (4)  

Natural language

AND (&) is true when both constituent propositions are true and false in all other circumstances viz. false false (¬P & ¬Q / 0 0 ), true false (P & ¬Q / 1 0 ), false true (¬P & Q / 0 1 )

AND at 0 0

AND at 1 0 or 0 1

Symbol for AND gate

It's very similar to NAND so be careful not to confuse it

OR

OR (in logic known as disjunction) in its non-exclusive form is true if either of its propositions are true or both are true . It is false otherwise.

p    q   p V q
_    _   _____

t    t     t      (1)
t    f     t      (2)
f    t     t      (3)
f    f     f      (4)  

XOR

XOR stands for exclusive or, also known as exclusive conjunction. This means it can only be true if one of its propositions are true . If both are true this doesn't exclude one of the propositions so the overall statement has to be false . This is the only change in the truth conditions from OR .

Electrical symbol for XOR

p    q   p X V q
_    _   ________

t    t     f      (1)
t    f     t      (2)
f    t     t      (3)
f    f     f      (4)  

**NOR**

This is equivalent to saying 'neither' in natural language. It is only true both propositions are false . If either one of the propositions is true the outcome is false . If both are true it is false

XNOR

This one is confusing. I can see the truth conditions but don't understand them. It is true if both propositions are false like NOR or if both propositions are true and false otherwise.

p    q   p ¬V q
_    _   ________

t    t     f      (1)
t    f     f      (2)
f    t     f      (3)
f    f     t      (4)  

p    q   p X¬V q
_    _   ________

t    t     t      (1)
t    f     f      (2)
f    t     f      (3)
f    f     t      (4)