CHAPTER 2 :BOOLEAN OPERATIONS *

Course Learning Outcomes, CLO

CLO 2:simplify logical expressions by using Boolean algebra and related techniques orderly.

CLO 3:perform correctly basic arithmetic operations by using Boolean laws.

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SUMMARY

BOOLEAN OPERATIONS Symbols, truth table, logic gates applications; NOT, AND, OR, NOR, NAND, XOR, XNOR. Laws of Boolean Algebra, Sum of Product (SOP), Product of Sum (POS) and Karnaugh Map. *

TRUTH TABLESA truth table is a table that describes the behavior of a logic gateThe number of input combinations will equal 2N for an N-input truth table**

Circuits which perform logic functions are called gatesThe basic gates are:I. NOT/INVERTER gateII.AND gateIII. OR gateIV.NAND gateV.NOR gateVI.XOR gateVII.XNOR gate*

I.NOT/INVERTER gate

SymbolTruth TableTiming Diagram*

II.AND gate

SymbolTruth TableTiming Diagram*

III.OR gate

SymbolTruth TableTiming Diagram*

IV. NAND gate*

V. NOR gate

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VI.XOR gate

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VII. XNOR gate

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BOOLEAN ALGEBRAThe Boolean algebra is an algebra dealing with binary variables and logic operationThe variables are designated by:Letters of the alphabetThree basic logic operations AND, OR and NOTA Boolean function can be represented by using truth table. A truth table for a function is a list of all combinations of 1s and 0s that can be assigned to the binary variable and a list that shows the value of the function for each binary combinationA Boolean expression also can be transformed into a circuit diagram composed of logic gates that implements the function*

Examples F = A + BC

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BASIC IDENTITIES AND BOOLEAN LAWS*

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All these Boolean basic identities and Boolean Laws can be useful in simplifying a logic expression, in reducing the number of terms in the expressionThe reduced expression will produce a circuit that is less complex than the one that original expression would have produced.Examples Simplify this function F = A B C + A B C + A C

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STANDARD FORMProduct termA term with the product of literalsThe AND of literalsBoolean multiplication

Sum termA term with the sum of literalsThe OR of literalsBoolean addition

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Sum of Product (SOP)A SOP is a switching expressions consisting either of a single product term or the OR (sum) of product termStandard form of SOP is where all the variables in the domain appear in each product term in the expressionThere are two steps to convert the equation into a standard form of SOP:Multiply each of the nonstandard term with the missing term using Boolean algebra A + A = 1 2. Repeat until all variables appear in each product term.

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ExamplesConvert this Boolean equation into standard form of SOPF = A + B C + A B C

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Product of Sum (POS)POS is when two or more sum term is multiplied togetherStandard form of POS is where all the variables in the domain appear in each sum term in the expressionThere are three steps in a way to convert a product term to standard form of POS:Multiply each of the nonstandard term with the missing term using Boolean algebra A A = 0Second step: Apply the Boolean identities again (A + B C) = (A + B) (A + C)3. Third step: Repeat until all variables appear in each product term.

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Examples Convert this Boolean equation into standard form of POSF = (A + B) (A + B + C) (B + C)

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KARNAUGH MAPprovides a systematic method for simplifying a Boolean expression or a truth table functionThe K-map is a table consisting of N = 2n cells, where n is the number of input variables.The table format is such that there is a single variable change between any adjacent cells*

Two variables K-map with assume A and B as variableThree variable K-map with assume A, B and C as variable*

Looping Looping of pairLooping a pair of adjacent 1s in a K-map eliminates the variable that appears in complemented and uncomplemented form*

Looping group of fourLooping a quad of adjacent 1s eliminate the two variables that appear in both complemented and uncomplemented form*

Looping group of eightLooping an octet of adjacent 1s eliminates the three variables that appear in the both complemented and uncomplemented form*

Examples Simplify this Boolean equation by using K-mapsF = X Y Z+ X Y Z + X Y Z

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ExamplesFind out the equation based on the given*

Solution

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