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General Information, Facts & Tables
[Boolean Mathematics]

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Overview

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Boolean logic is the basis of digital electronics and computing. By combining these simple gates together, extremely complex algorithms can be calculated.
 

Basic Gates

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The three basic gates are AND, OR and NOT, which can be combined in various configurations to create all other gates. There is a direct relationship of logic gates to electronics, and some of the gates can be easily explained as simple electric circuit diagrams.
  

AND Gate

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Symbol
AND Gate Symbol

AND Truth Table

In A

In B

Out C

True (1)

True (1)

True (1)

True (1)

False (0)

False (0)

False

True (1)

False (0)

False (0)

False (0)

False (0)

The AND gate could be thought of as two switches in series.
 

Notation
AND Gate Notation

Electronic Representation
AND Gate Electronic Representation

OR Gate

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Symbol
OR Gate Symbol

OR Truth Table

In A

In B

Out C

True (1)

True (1)

True (1)

True (1)

False (0)

True (1)

False (0)

True (1)

True (1)

False (0)

False (0)

False (0)

The OR gate could be thought of as two switches in parallel.
 

Notation
OR Gate Notation

Electronic Representation
OR Gate Electric Representation

NAND Gate

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Symbol
NAND Gate Symbol

NAND Truth Table

In A

In B

Out C

True (1)

True (1)

False (0)

True (1)

False (0)

True (1)

False (0)

True (1)

True (1)

False (0)

False (0)

True (1)

Also known as a NOT AND gate.
 

Notation
NAND Gate Notation

NOR Gate

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Symbol
NOR Gate Symbol

NOR Truth Table

In A

In B

Out C

True (1)

True (1)

False (0)

True (1)

False (0)

False (0)

False (0)

True (1)

False (0)

False (0)

False (0)

True (1)

Also known as a NOT OR gate.
 

Notation
NOR Gate Notation

XOR Gate

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Symbol
XOR Gate Symbol

XOR Truth Table

In A

In B

Out C

True (1)

True (1)

False (0)

True (1)

False (0)

True (1)

False (0)

True (1)

True (1)

False (0)

False (0)

False (0)

Also known as an Exclusive OR gate.
 

Notation
XOR Gate Notation

XNOR Gate

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Symbol
XNOR Gate Symbol

Truth Table

In A

In B

Out C

True (1)

True (1)

True (1)

True (1)

False (0)

False (0)

False (0)

True (1)

False (0)

False (0)

False (0)

True (1)

Also known as an Exclusive NOT OR gate.
 

Notation
XNOR Gate Notation

NOT (Inverter) Gate

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Symbol
NOT Gate Symbol

NOT Truth Table

In A

Out B

True (1)

False (0)

False (0)

True (1)

The  NOT gate reverses (inverts) the state of the input signal.  

Notation
NOT Gate Notation

Buffer Gate

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Symbol
Buffer Gate Symbol

Buffer Truth Table

In A

Out B

True (1)

True (1)

False (0)

False (0)

The buffer gate is used in electronics to ensure that the voltage of the logic output signal is the correct voltage. It is not used in other representations of logic. It is included here for completeness only.
 

Notation
Buffer Gate Notation

Boolean Algebra Theorems

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Commutative Property

AB = BA

A+B = B+A

Associative Property

A(BC) = (AB)C

A+(B+C) = (A+B)+C

Distributive Property

A(B+C) = AB + AC

A+(BC) = A+B(A+C)

Theorems

A AND 0 = 0

A AND 1 = A

A AND A = A

A AND A = 0

A = NOT A

A OR 0 = A

A OR 1 = 1

A OR A = A

A OR NOT A = 1

DeMorgans Theorem

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DeMorgans Theorem states that "Any logical function can be implemented using either (inverters and AND gates) or (inverters and OR gates).
DeMorgan's Identities

NOT(A OR B) = NOT(A) AND NOT(B)
NOT(A AND B) = NOT(A) AND NOT(B)

1

2

3

4

5

6

7

8

9

10

A

B

NOT A

NOT B

A AND B

NOT(A AND B)

A OR B

NOT (A OR B)

NOT(A) OR NOT(B)

NOT(A) NAND NOT(B)

F (0)

F (0)

T (1)

T (1)

F (0)

T (1)

F (0)

T (1)

T (1)

T (1)

F (0)

T (1)

T (1)

F (0)

F (0)

F (0)

T (1)

F (0)

F (0)

T (1)

T (1)

F (0)

F (0)

T (1)

F (0)

F (0)

T (1)

F (0)

F (0)

T (1)

T (1)

T (1)

F (0)

F (0)

T (1)

F (0)

T (1)

F (0)

F (0)

F (0)

 

Notes

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The logic gates described above have only one or two inputs. With the exception of "Inverters" (NOT) and "Buffers" all other gates can accept as many inputs as required.