# General Information, Facts & Tables [Boolean Mathematics]

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

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 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

Electronic Representation

OR Gate

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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

Electronic Representation

NAND Gate

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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

NOR Gate

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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

XOR Gate

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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

XNOR Gate

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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

NOT (Inverter) Gate

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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

Buffer Gate

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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

Boolean Algebra Theorems

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 Commutative Property Associative Property Distributive Property Theorems

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
 1 2 3 4 5 6 7 8 9 10 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.