15. BOOLEANALGEBRA

8/7/2019 15. BOOLEANALGEBRA

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LOGIC GATEFUNCTION


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Two ways to describe thefunction of Logic Gate

Truth TableBOOLEANALGEBRA


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Truth Table1. Truth Table - is a tabulated list of allpossible input and output combinations of alogic device.


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Understanding Truth Tables

Truth tables help understand thebehavior of logic gates.

y show how the input(s) of a logic gaterelate to its output(s).

y The gate input(s) are shown in the leftcolumn(s) of the table with all thedifferent possible input combinations.This is normally done by making theinputs count up in binary.

y The gate output(s) are shown in the

right hand side column.


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Example:

Given the following statement, identify thevariables and assign a value to each:

The president of a company and 3assistants are voting on whether to accept acontract. If the president votes for it, then 2yes votes (including the presidents) areenough; but if the president votes no, all 3assistants must vote yes in order for thecontract to be accepted.


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24 = 16 lines in thetruth table

P A1 A2 A3 C

0 0 0 0 0

0 0 0 1 0

0 0 1 0 0

0 0 1 1 0

0 1 0 0 00 1 0 1 0

0 1 1 0 0

0 1 1 1 1

1 0 0 0 0

1 0 0 1 1

1 0 1 0 1

1 0 1 1 1

1 1 0 0 1

1 1 0 1 1

1 1 1 0 1

1 1 1 1 1


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U nfortunately, they may quickly becamelarge and very long process.Ex. For 8 inputs variable: 2 8 = 256 entries.


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LOGIC GATES

Basic building blocks in digital electronics.Internally composed of an electronic circuitusually transistor-based to provide a presetlogic function.Designed to provide a logical high or lowoutput depending on the condition of theinputs.


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NOT FUNCTION

It has one input and one output.

To complement the logic signalat its input.


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NOT GATE or INVERTER


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OR GATE FUNCTION

Provides a high or 1 output

when at least one of its inputs is1Output is low or 0 when allinputs are 0May have two or more inputs.


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OR GATE/TRUTH TABLE


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AND GATE FUNCTION

When one or more inputs are

low, the output is also low.The output goes high only whenall inputs are high.


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AND GATE/TRUTH TABLE


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NOR GATE FUNCTION

Derived from NOT and OR gate

It produces a high logic outputwhen the inputs are all at logiclow level, and a low logicoutput when at least one input isat a high logic level.


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NAND GATE FUNCTION

Produces a low logic output only whenall its inputs are highThis is simply complement of the ANDgate

When one or more inputs are low, theoutput is highThe output goes low only when all

inputs are high.


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NAND GATE/TRUTH TABLE


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X OR GATE FUNCTION

Produces a high logic output

only when one but all its inputsis high.When the inputs are all high or low the output is low.


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X NOR GATE FUNCTION

A complement of the exclusive

OR gate.It produces a low logic outputonly when but not all its inputsis high.


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X NOR GATE/TRUTH TABLE


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2. BOOLEAN ALGEBRA

2. BOOLEANALGEBRA - is a branch of mathematics that is directly applicable to

digital design because equations of Booleanalgebra can be physically implemented byusing electronics gates.This system offers a mathematical approachto representing the logic function: AND,OR, NOT gate particularly whensimplifying complex combinations.


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Addition The OR Operation

The OR gate is an electronic circuitthat gives a high output (1) if one ormore of its inputs are high. A plus

(+) is used to show the OR operation.


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Multiplication The ANDOperation

The AND gate is an electronic circuit thatgives a high output (1) only if all itsinputs are high. A dot (.) is used to show

the AND operation i.e. A.B. Bear inmind that this dot is sometimes omittedi.e. AB


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Other Permissible BooleanOperations

1. Factoring Ex. xy + xz = x(y + z)

Prohibited Operations:

1.Subtraction2.Division


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Logical Manipulation

In the practical world, each logic expressionis translated into electronics gates.

Therefore, the simplest and most compactBoolean statement results in the fewestgates and wires.

Boolean Theorems are used to simplifyexpressions and equations designs.


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BOOLEAN THEOREMS:

1. x + 0 = x2. x + 1 = 1

3. x . 0 = 04. x . 1 = X5 . x . x = x

6. x + x = x7. x . x = 08. x + x = 1

9. x + xy = x10. x[x + y] = x

11. x + xy = x + y12. xy + yz + xz =

xy + yz

13. [x + y][x + y] = x14. [y + z][y + x] =xy + yz


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EX AMPLES

1. Do not simplify the followingexpressions before implementing.

2. Simplify first, then implement theexpressions.[x + x + y] xz + xz[y + y]

xy + xy + xz[w + x][x + y][w + x + y + z] + x + y +w


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Simplify and implement the followinglogic expressions:

1. x + x + y2. x[y + z] x + w3. x[y + z] + yz + xz4. [A + B][A + BC] + AB + AC5 . wx + wxy + wxz + wy


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1. AB+ AB + AB2. AB + AB + AB3. AB+ AB + AB + AB4. ABC+ ABC5 . ABC + ABC + ABC + ABC6. ABC + ABC + ABC + ABC7. ABC + ABC + ABC + ABC + ABC8. ABC + ABCD + ABCD9. ABD + ABCD + ABCD + ABCD + ABCD

+ ABCD10. ABC + ABC + ABC + ABC + ABC + ABC

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15. BOOLEANALGEBRA