# Fundamental Hardware Elements of Computers

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• Created by: Steph16
• Created on: 10-04-15 13:29
• Fundamental Hardware Elements of Computers
• Logic Gates
• NOT
• Used to change a condition
• AND
• Used to check that both of two conditions are true
• OR
• Used to see if either of two conditions are true
• NAND
• Provides the inverse of the AND operator
• NOR
• Provides the inverse of the OR operator
• XOR
• Used to see if only one or the other applies
• Usually used in conjunction with relational operators, which return a True/False value.
• Logic Diagrams
• A way of visualising a logical procedure.
• AND
• a.b
• OR
• a+b
• NAND
• (a.b)'
• NOR
• (a+b)'
• XOR
• (a.b) + (a.b)
• NOT
• a'
• Boolean Circuits
• The information flows from left to right
• Logic gates symbols can be combined to create circuits of logical operations that, when combined, process the input to give an output.
• Boolean Algebra
• A variable can either be 1 or 0
• 0+0 = 0
• 1+1= 1
• 0.0 = 0
• 1.1 = 1
• 0.1 = 1.0 = 0
• 1 + 0 = 0 + 1 = 1
• Boolean Algebra Law
• A + A = A
• A . A = A
• Identity law. Adding or multiplying two true outcomes will always result in another true
• A + A = A
• A + (B.C) = (A+B). (A+C)
• A . (B + C) = (A.B) + (A.C)
• This means that Boolean algebra is distributive so a multiplication can always be rewritten as the sum of two other maltiplications
• A + (B.C) = (A+B). (A+C)
• A + (A.B) = A
• A. (A+B) = A
• Redundancy law. A + (A.B) = A. (1 + B) = A. 1 = A
• A + (A.B) = A
• 1 + A = 1
• 1 . A = A
• If there is one part true in a Boolean expression then the outcome will be true, so the effect of multiplying or adding with a true value means that the result will always be true
• 1 + A = 1
• A.B = B.A
• This means that Boolean algebra is commutative; the order of the inputs is not important
• A + B = B + A
• Boolean Algebra Law
• A . A = A
• Identity law. Adding or multiplying two true outcomes will always result in another true
• A . (B + C) = (A.B) + (A.C)
• This means that Boolean algebra is distributive so a multiplication can always be rewritten as the sum of two other maltiplications
• A. (A+B) = A
• Redundancy law. A + (A.B) = A. (1 + B) = A. 1 = A
• 1 . A = A
• If there is one part true in a Boolean expression then the outcome will be true, so the effect of multiplying or adding with a true value means that the result will always be true
• A.B = B.A
• This means that Boolean algebra is commutative; the order of the inputs is not important
• A + B = B + A
• Boolean Algebra Law
• 0 + A = A
• 0.A = 0
• This law simply describes the effect of adding or multiplying by 0
• 0 + A = A
• A + /A\ = 1
• One of them will always be true (1) and one of them will always be false (0)
• A . /A\ = 0
• A . /A\ = 0
• A.B + A./B\ = A
• (A + B) . (A + /B\) = A
• Because either B or not B will be true multiplying by each of them will always result in one of the sums having a true outcome.
• A.B + A./B\ = A
• A + (/A\ . B) = A+B
• A and /A\ will always have one true and one false result
• A . (/A\ + B) = A. B
• A . (/A\ + B) = A. B
• (A+B) + C = A + (B+C)
• This means that the Boolean algebra is associative; the order that the sum is calculated is not important
• (A . B) . C =  A . (B . C)
• This means that the Boolean algebra is associative; the order that the sum is calculated is not important
• De Morgan's Law
• /(A + B)\ = /A\ . /B\
• /(A . B)\ = /A\ + /B\
• This shows that the following 2 sets of circuits were equivalent. Simplifying a 3 gate circuit with the simpler two gate circuit
• Using this law and all of the other Boolean laws you are able to simplify the circuits and hence more understandable, and more importantly take up less space if they are used in hardware devices
• Key: / \ shows a line above the letter, to signify that the letter is NOT.  + means OR.  (.) means AND.