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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

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Page 1: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 2: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 3: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 4: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 5: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 6: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 7: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

SOP (cont.)

DCABDCBA

DCBADCBACDBADCBACDBA

DCABDDCCBADDCBA ))(()(

DCABDDCBACBADCBACDBA ))((

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

Page 8: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 9: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 10: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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?

))()(( DCBADCBCBA

=> No

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

Page 11: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

))()(( DCBADCBCBA

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

Page 12: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

POS (cont.)

))()(( DCBADCBCBA

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

))(.)(.( DCBADCBAADDCBA

Expand first and second terms

)(

))()()((

DCBA

DCBADCBADCBADCBA

Standard POS form

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

Page 13: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 14: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 15: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 16: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 17: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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

Page 18: Logic Gates and Boolean Algebra€¦ · Logic Gates and Boolean Algebra •Logic Gates –Inverter, OR, AND, Buffer, NOR, NAND, XOR, XNOR •Boolean Theorem –Commutative, Associative,

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