Boolean Algebra - FA Revision

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

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

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) 

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

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

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

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

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