     # Logic Gates and Boolean Algebra - Logic Gates and Boolean Algebra â€¢Logic Gates â€“Inverter, OR, AND,

• View
1

0

Embed Size (px)

### Text of Logic Gates and Boolean Algebra - Logic Gates and Boolean Algebra â€¢Logic Gates...

• Logic Gates and Boolean Algebra

• Logic Gates – Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR

• Boolean Theorem – Commutative, Associative, Distributive Laws – Basic Rules

• DeMorgan’s Theorem • Universal Gates

– NAND and NOR

• Canonical/Standard Forms of Logic – Sum of Product (SOP) – Product of Sum (POS) – Minterm and Maxterm

2/18/2012 1 A.A.H Ab-Rahman, Z.Md-Yusof

• SOP and POS

• All boolean expressions can be converted to two standard forms:

– SOP: Sum of Product

– POS: Product of Sum

• Standardization of boolean expression makes evaluation, simplification, and implementation of boolean expressions more systematic and easier

2/18/2012 2 A.A.H Ab-Rahman, Z.Md-Yusof

• Sum of Product (SOP)

• Boolean expressions are expressed as the sum of product, example:

• Each variable or their complements is called literals

• Each product term is called minterm

DCBCDEABC  literal

minterm

2/18/2012 3 A.A.H Ab-Rahman, Z.Md-Yusof

• SOP (cont.)

• In SOP, a single overbar cannot extend over more than one variable, example:

• Standard SOP forms must contain all of the variables in the domain of the expression for each product term, example:

BCAAB  Not SOP because BC

ABCCBACBA 

2/18/2012 4 A.A.H Ab-Rahman, Z.Md-Yusof

• SOP (cont.)

• In the following SOP form,

– How many minterms are there?

– How many literals in the second product term?

– Is it in a standard SOP form?

– How do we convert the boolean expression to standard SOP form?

DCABBACBA 

=> 3

=> 2

=> No

2/18/2012 5 A.A.H Ab-Rahman, Z.Md-Yusof

• SOP (cont.)

• To convert SOP to its standard form, we use the boolean rules

– A + A = 1

– A(B + C) = AB + AC

• We have

• The first product term is missing the variable D, and the second product term is missing C and D

DCABBACBA 

2/18/2012 6 A.A.H Ab-Rahman, Z.Md-Yusof

• SOP (cont.)

DCABDCBA



DCABBACBA 

Apply D + D = 1 and C + C = 1

Apply the distributive law

Standard SOP form

2/18/2012 7 A.A.H Ab-Rahman, Z.Md-Yusof

• Product of Sum (POS)

• Boolean expressions are expressed as the product of sum, example:

))(( CBABA  literal

maxterm

2/18/2012 8 A.A.H Ab-Rahman, Z.Md-Yusof

• POS (cont.)

• In POS, a single overbar cannot extend over more than one variable, example:

• Standard POS forms must contain all of the variables in the domain of the expression for each sum term, example:

Not SOP because B+C ))(( CBABA 

))()(( CBACBACBA 

2/18/2012 9 A.A.H Ab-Rahman, Z.Md-Yusof

• POS (cont.)

• In the following POS form,

– Is it in a standard POS form?

– How do we convert the boolean expression to standard POS form?

=> No

2/18/2012 10 A.A.H Ab-Rahman, Z.Md-Yusof

• POS (cont.)

• To convert POS to its standard form, we use the boolean rules

– A . A = 0

– A + BC = (A + B)(A + C)

• We have

• The first sum term is missing the variable D, and the second sum term is missing A

2/18/2012 11 A.A.H Ab-Rahman, Z.Md-Yusof

• POS (cont.)

Apply D.D = 0 and A.A = 0 to first and second terms

Expand first and second terms

)(

))()()((

DCBA





Standard POS form

2/18/2012 12 A.A.H Ab-Rahman, Z.Md-Yusof

• Minterm and Maxterm • Minterm: Product terms in SOP

• Maxterm: Sum terms in POS

• Standard forms of SOP and POS can be derived from truth tables

A B C Z

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

CBA 

CBA

CBA 

CBA 

CBA 

CBA

CAB

ABC

ABCCABCBACBAZ 

))(( CBACBAZ 

For SOP form,

For POS form,

 )7,6,5,1(m

 )4,3,2,0(M ))(( CBACBA 

2/18/2012 13 A.A.H Ab-Rahman, Z.Md-Yusof

• Minterm and Maxterm

• How to design minterms – AND-OR logic

ABCCABCBACBAZ 

A B C

A B C

A B C

A B C

Z

Also known as

2 level logic

2/18/2012 14 A.A.H Ab-Rahman, Z.Md-Yusof

• Minterm and Maxterm

• How to design minterms – NAND-NAND Logic

A B C

A B C

A B C

A B C

Z

SRQPZ 

P

Q

R

S

Using DeMorgan’s Theorem

SRQPZ 

ABCCABCBACBAZ 

2/18/2012 15 A.A.H Ab-Rahman, Z.Md-Yusof

• Minterm and Maxterm

• How to design maxterms – OR-AND Logic

))()()(( CBACBACBACBAZ 

A B C

A B C

A B C

A B C

Z

2/18/2012 16 A.A.H Ab-Rahman, Z.Md-Yusof

• Minterm and Maxterm

• How to design maxterms – NOR-NOR Logic

A B C

A B C

A B C

A B C

Z

))()()(( CBACBACBACBAZ 

P

Q

R

S SRQPZ 

Using DeMorgan’s Theorem

SRQPZ  2/18/2012 17 A.A.H Ab-Rahman, Z.Md-Yusof

• Minterm and Maxterm

• Can the minterm and maxterm logic be optimized?

– Yes, using Boolean algebra – explore yourself

– Yes, using Karnaugh maps – next lecture

A.A.H Ab-Rahman August 2008 2/18/2012 18 A.A.H Ab-Rahman, Z.Md-Yusof

Recommended ##### Introduction to Computing Systems from bits & gates to C & beyond Chapter 3 Digital Logic Structures Transistors Logic gates & Boolean logic Combinational
Documents ##### Chapter 2 Boolean Algebra and Logic Gates. 2 Chapter 2. Boolean Algebra and Logic Gates 2-2Basic Definitions 2-3AxiomaticDefinition of Boolean Algebra
Documents ##### Binary Logic, Boolean Algebra and digital logic gates, ... design of logic circuits using tools such
Documents ##### Boolean Algebra and Logic Gates - Boolean algebra (i.e., a Boolean algebra with only two elements)
Documents ##### Digital Logic: Boolean Algebra and Gates - Courses Ttb kCh t 3 Digital Logic: Boolean Algebra and Gates Textbook Chapter 3 Basic Logic Gates CMPE12 – Summer 2009 02-2 XOR
Documents ##### 1 Boolean Algebra and Logic Gates EE 240 – Logic Design Chapter 2 Sohaib Majzoub
Documents ##### boolean algebra and logic gates-CCupload Boolean Algebra and Logic Gates 0908531 Mechatronics System
Documents ##### Chapter 4 Logic Gates and Boolean Algebra. Introduction Logic gates are the actual physical implementations of the logical operators. These gates form
Documents Documents