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# Logical operations & boolean algebra

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2013

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### Logical operations & boolean algebra

1. 1. LOGICAL OPERATIONS& BOOLEAN ALGEBRAANDREI JECHIU
2. 2. BOOLEAN ALGEBRABoolean algebra derives its name from the mathematician GeorgeBoole.A Boolean algebra value can be either true or false.˗True is represented by the value 1.˗False is represented by the value 0.
3. 3. ANDOutput is one if every input has value of 1More than two values can be “and-ed” togetherFor example xyz = 1 only if x=1, y=1 and z=1x y out = xy0 0 00 1 01 0 01 1 1xyout
4. 4. OROutput is 1 if at least one input is 1.More than two values can be “or-ed” together.For example x+y+z = 1 if at least one of the three values is 1.x y out = x+y0 0 00 1 11 0 11 1 1xyout
5. 5. NOTThis function operates on a single Boolean value.Its output is the complement of its input.An input of 1 produces an output of 0 and an input of 0 produces anoutput of 1x xx x0 11 0
6. 6. XOR (EXCLUSIVE OR)The number of inputs that are 1 matter.More than two values can be “xor-ed” together.General rule: the output is equal to 1 if an odd number of input valuesare 1 and 0 if an even number of input values are 1.x y out =0 0 00 1 11 0 11 1 0yxxyout
7. 7. NANDOutput value is the complemented output from an “AND” function.x y out = x NAND y0 0 10 1 11 0 11 1 0xyout
8. 8. NORx y out = x NOR y0 0 10 1 01 0 01 1 0xyoutOutput value is the complemented output from an “OR” function.
9. 9. XNOROutput value is the complemented output from an “XOR” function.xyoutx y out =x xnor y0 0 10 1 01 0 01 1 1
10. 10. Identity name AND form OR formIdentity Law x1 = x x + 0 = xNull (or Dominance) Law 0x = 0 1+x = 1Idempotent Law xx = x x+x = xInverse LawCommutative Law xy = yx x+y = y+xAssociative Law (xy)z = x(yz) (x+y)+z =x+(y+z)Distributive Law x + y z = (x + y) (x + z) x(y + z) = xy+xzAbsorption Law x(x+y) = x x+xy = xDeMorgan’s LawDouble Complement LawBASIC BOOLEAN IDENTITIES
11. 11. EXAMPLEInputs Intermediates Outputx y a b z0 0 0 1 10 1 0 0 01 0 0 0 01 1 1 0 1
12. 12. RESOURCES• Boolean algebra• Boolean algebra laws• Boolean algebra #1: Basic laws and rules• Boolean Algebra Examples