Electronic sdevices recognise the presence of a current or the absence of a current

This is recognised either with a 1 or a 0

Computers are comprised of billions of switches which can either be ON or OFF

These switches can be combined in different ways to create simple circuits known as logic gates

Logic gates can take multiple inputs to produce a single output

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Logic Gates

Logic Gates:

Electronic logic gates can take one or more inputs to produce a single output

The output can then become the input to the next gate and so on to create a complex circuit

A number of logic gates are designed to produce different outputs for the various possible combinations of ON or OFF inputs

Inputs and outputs of each logic gate are represented by Truth Tables

Truth Tables are simple diagrams which quickly record the function of logic gates

Individual logic gates can quickly be calculated however complex circuits where outputs need to be quickly understood benefit from Truth Tables

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De Morgan's First Law

De Morgan's First Law:

De Morgan's First Law states that ¬(AvB) = ¬A^¬B

Using the Venn diagram, the white area represents A OR B (AvB)

X represents all of the blue area - NOT (A OR B) (¬(AvB))

The blue area is everything that is (NOT A) AND (NOT B) (¬ A^¬B)

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De Morgan's Second Law

De Morgan's Second Law:

De Morgan's Second Law states that ¬(A^B) = ¬Av¬B

Looking at the Venn diagram, if X=¬(A^B), X cannot be in the centre so it must be everywhere else

This means that X is either not in A, not in B, or not in either

This is the definition of X=¬Av¬B

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Absorption Law

Absorption Law:

Absorbption law states that in a complicated expression it is possible to simplify an expression into a simpler expression by absorbing like terms

This allows expressions to be simplified or reduced into a more simple expression

Absorption law states that A+AB=A is true and and can be simplified to A(A+B)=A

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Karnaugh Maps

Karnaugh Maps:

Karnaugh Maps are used as truth tables for complex boolean expressions while providing an alternative and often easier method of simplifying expressions

Karnaugh Maps use the Absorption law in order to represent complex boolean expressions in their simplest form

Typically, when groups are formed through absorption in Karnaugh Maps, they represent a more complex expression than stated by the headers of each row and column within the map

In the example overleaf, all the squares where A is true are filled in

The all the squares where A^B are true are filled in

The adjacent 1's are grouped together and the expression A^B is represented by the A group as the A^B expression has been absorbed

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Half Adder Circuits

Half Adder Circuits:

A half adder circuit performs the addition two bits

It takes an input of two bits (A and B) and outputs the Sum (S) and the Carry (C)

S represents the sum S=AvB

C represents the carry C=A^B

The half adder only has two inputs so it cannot use the carry from a previous addition as a third input to a subsequent addition

A half adder can only add one bit numbers

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Full Adder Circuits

Full Adder Circuits:

A Full Adder Circuit is comprised of two half adder circuits

A Full Adder has 3 inputs (A, B, Carry (Cin)) and two outputs (S and Carry (Cout))

The second half adder inputs the Carry (Cin) from the first operation

The second half adder outputs S and the new carry (Cout)

Full adders can be concatenated in order to perform operations with multiple bits and take multiple inputs as well as multiple carries

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