Boolean Logic and Algebra: Gates, Truth Tables, and Postulates, Slides of Digital Logic Design and Programming

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Binary (Boolean) Logic

 Deals with binary variables and binary logic

functions

 Has two discrete values

 0  False, Open

 1  True, Close

 Three basic logical operations

 AND (.); OR (+); NOT (‘)

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Logic Gates & Truth Tables

AND OR^

NOT

A

OR

B

A+B

A

A

AND

B

A.B

A’

NOT

 AND; OR gates may have any # of inputs

 AND  1 if all inputs are 1; 0 other wise

 OR  1 if any input is 1; 0 other wise

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Boolean Algebra - Postulates

3. Postulate 3:

Commutative with respect to + and.

4. Postulate 4:

Distributive over. and +

5. Postulate 5:

For each element a of B, there exist an element a’such that (a) a + a’= 1 and (b) a.a’= 0

6. Postulate 6:

There exists at least two elements a, b in B, such that a ≠b

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Boolean Algebra - Postulates

 Postulates are facts that can be taken as true; they do

not require proof

 We can show logic gates satisfy all the postulates 1 0 0 1 0 1 1

A

OR

B

A+B

A

A

AND

B

A.B

A’

NOT

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Boolean Functions

 Are represented as

 Algebraic expressions; F1 = x + y’z  Truth Table

 Synthesis

 Realization of schematic from the expression/truth table

 Analysis

 Vice-versa

x 1 0 0 1

y 1

z 1

F

x

y

z

F

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Synthesis – F

 Assume true as well as complement inputs

are available

 Cost

 A 2-input AND gate  A 2-input OR gate  4 inputs 1 0 0^1 1 0 1 1

x 1 0 0 1

y 1

z 1

F

x

y

z

F

11 Minterms  One method of Writing Boolean function is the canonical minterm (sum of products or SOP) form  F = x’y’z +xy’z + xyz’= m1 + m5 + m6 = ∑(1,5,6) xyz xyz’ xy’z xy’z’ x’yz x’yz’ x’y’z x’y’z’ Corresponding minterm 1 0 0 m 4 1 0 1 m 5 0 1 1 m 3 1 1 0 m 6 1

x 1 0 m 2 1

y 1

z m 7 m 1 m 0 Designation

12 Minterms – examples F2 = ∑(0,1,2,3,5) = x’y’z’+ x’y’z + x’yz’+ x’yz + xy’z

F2 (Given) 1 0 0 1 0 1 m 5 0 1 1 m 3 1 1 0 1

x 1 0 m 2 1

y 1

z m 1 m 0 Designation

14 Maxterms  A maxterm is an OR term in which every literal (variable) or its complement in a function occurs once  Each maxterm has a value 0 for one combination of values of n variables x’+y’+z’ x’+y’+z x’+y +z’ x’+y +z x +y’+z’ x +y’+z x +y +z’ x +y +z Corresponding maxterm 1 0 0 M 4 1 0 1 M 5

0 1 1 M 3

1 1 0 M 6

x 1 0 M 2 1

y 1

z M 7

M 1

M 0

Designation

15 Minterms & Maxterms  Conversion between minterms & maxterms m 0 = x’y’z’= (x+y+z)’= (M 0 )’ In general, mi = (Mi)’  An alternative method of writing a Boolean function is the canonical maxterm (product of sums or POS) form  The canonical product of sums can be written directly from the truth table

17 Standard Forms  In canonical forms, each minterm (or maxterm) must contain all variables (or its complements) The algebraic expressions can further be simplified  Example F4 (x,y,z) = xy +y’z (sum of products, standard form) F5 (x,y,z) = (x+y’)(y+z) (product of sums, standard form)  Conversion

Standard form can be converted into canonical form using identity elements

F4 = xy + y’z = xy.1 +1.y’z = xy(z+z’) + (x+x’)y’z = xyz + xyz’+ xy’z + x’y’z = m7 +m6 +m5 +m  How about the conversion from canonical forms to standard forms?  Exercise – convert F5 into maxterms

18 Non-Standard Forms  A Boolean function may be written in non-standard form F6 (x,y,z) = (xy + z)(xz + y’z) = xy(xz + y’z) + z(xz + y’z) = xyz + xyy’z + xz +y’z = xyz + xz + y’z = xz + y’z (standard form)

20 Other Logic Gates – NOR Gate  2 - input NOR (NOT-OR operation)  Can have any # of inputs  NOR gate is not associative

Associative property to be discussed later

x

y

z

x 0 0

y 0

z

21 Other Logic Gates – XOR Gate  2 - input XOR  Output is 1 if any input is one and the other input is 0  Can have any # of inputs

x

y

z

x 0 1

y 1

z