Boolean Algebra - FA Revision
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- Created by: ac.x123
- Created on: 11-05-22 15:59
Binary Switches
Binary Switches:
- 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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