
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 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.
