# Lecture 5: More Logic Functions: NAND, NOR, XOR

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Lecture 5: More Logic Functions: NAND, NOR, XOR

Syed M. Mahmud, Ph.D ECE Department

Wayne State University

Original Source: Prof. Russell Tessier

ENGIN112 L7: More Logic Functions

September 17, 2003

Overview

° More 2-input logic gates (NAND, NOR, XOR) ° Extensions to 3-input gates ° Converting between sum-of-products and NANDs

• SOP to NANDs • NANDs to SOP

° Converting between sum-of-products and NORs

• SOP to NORs • NORs to SOP

° Positive and negative logic

• We use primarily positive logic in this course.

ENGIN112 L7: More Logic Functions

September 17, 2003

Logic functions of N variables

° Each truth table represents one possible function (e.g. AND, OR)

N

° If there are NNinputs, there are 22 possible functions (truth tables).

° For example, is N is 2 then there are 16 possible truth tables.

° So far, we have defined 2 of these functions

• 14 more are possible.

° Why consider new functions?

• Cheaper hardware, more flexibility.

xyG 000 010 100 111

ENGIN112 L7: More Logic Functions

September 17, 2003

The NAND Gate AY B

° This is a NAND gate. It is a combination of an AND gate followed by an inverter. Its truth table shows this…

° NAND gates have several interesting properties…

• NAND(a,a)=(aa)’ = a’ = NOT(a)

• NAND’(a,b)=(ab)’’ = ab = AND(a,b) • NAND(a’,b’)=(a’b’)’ = a+b = OR(a,b)

ABY 001

011

101

110

ENGIN112 L7: More Logic Functions

September 17, 2003

The NAND Gate

° These three properties show that a NAND gate with both of its inputs driven by the same signal is equivalent to a NOT gate

° A NAND gate whose output is complemented is equivalent to an AND gate, and a NAND gate with complemented inputs acts as an OR gate.

° Therefore, we can use a NAND gate to implement all three of the elementary operators (AND,OR,NOT).

° Therefore, ANY switching function can be constructed using only NAND gates. Such a gate is said to be primitive or functionally complete.

ENGIN112 L7: More Logic Functions

September 17, 2003

NAND Gates into Other Gates

(what are these circuits?)

AY

NOT Gate

A

B

Y

A

AND Gate

Y B

ENGIN112 L7: More Logic Functions

OR Gate

September 17, 2003

The NOR Gate

AY B

° This is a NOR gate. It is a combination of an OR gate followed by an inverter. It’s truth table shows this…

° NOR gates also have several

interesting properties…

• NOR(a,a)=(a+a)’ = a’ = NOT(a) • NOR’(a,b)=(a+b)’’ = a+b = OR(a,b) • NOR(a’,b’)=(a’+b’)’ = ab = AND(a,b)

ABY 001 010 100

110

ENGIN112 L7: More Logic Functions

September 17, 2003

Functionally Complete Gates

° Just like the NAND gate, the NOR gate is functionally complete…any logic function can be implemented using just NOR gates.

° Both NAND and NOR gates are very valuable as any design can be realized using either one.

° It is easier to build an IC chip using all NAND or NOR gates than to combine AND,OR, and NOT gates.

° NAND/NOR gates are typically faster at switching and cheaper to produce.

ENGIN112 L7: More Logic Functions

September 17, 2003

NOR Gates into Other Gates

(what are these circuits?)

AY

NOT Gate

A

B

Y

A

OR Gate

Y B

ENGIN112 L7: More Logic Functions

AND Gate

September 17, 2003

The XOR Gate (Exclusive-OR)

A B

° This is a XOR gate. ° XOR gates assert their output

when exactly one of the inputs is asserted, hence the name. ° The switching algebra symbol for this operation is , i.e. 1 1 = 0 and 1 0 = 1.

ENGIN112 L7: More Logic Functions

Y

ABY 000 011 101 110

September 17, 2003

Syed M. Mahmud, Ph.D ECE Department

Wayne State University

Original Source: Prof. Russell Tessier

ENGIN112 L7: More Logic Functions

September 17, 2003

Overview

° More 2-input logic gates (NAND, NOR, XOR) ° Extensions to 3-input gates ° Converting between sum-of-products and NANDs

• SOP to NANDs • NANDs to SOP

° Converting between sum-of-products and NORs

• SOP to NORs • NORs to SOP

° Positive and negative logic

• We use primarily positive logic in this course.

ENGIN112 L7: More Logic Functions

September 17, 2003

Logic functions of N variables

° Each truth table represents one possible function (e.g. AND, OR)

N

° If there are NNinputs, there are 22 possible functions (truth tables).

° For example, is N is 2 then there are 16 possible truth tables.

° So far, we have defined 2 of these functions

• 14 more are possible.

° Why consider new functions?

• Cheaper hardware, more flexibility.

xyG 000 010 100 111

ENGIN112 L7: More Logic Functions

September 17, 2003

The NAND Gate AY B

° This is a NAND gate. It is a combination of an AND gate followed by an inverter. Its truth table shows this…

° NAND gates have several interesting properties…

• NAND(a,a)=(aa)’ = a’ = NOT(a)

• NAND’(a,b)=(ab)’’ = ab = AND(a,b) • NAND(a’,b’)=(a’b’)’ = a+b = OR(a,b)

ABY 001

011

101

110

ENGIN112 L7: More Logic Functions

September 17, 2003

The NAND Gate

° These three properties show that a NAND gate with both of its inputs driven by the same signal is equivalent to a NOT gate

° A NAND gate whose output is complemented is equivalent to an AND gate, and a NAND gate with complemented inputs acts as an OR gate.

° Therefore, we can use a NAND gate to implement all three of the elementary operators (AND,OR,NOT).

° Therefore, ANY switching function can be constructed using only NAND gates. Such a gate is said to be primitive or functionally complete.

ENGIN112 L7: More Logic Functions

September 17, 2003

NAND Gates into Other Gates

(what are these circuits?)

AY

NOT Gate

A

B

Y

A

AND Gate

Y B

ENGIN112 L7: More Logic Functions

OR Gate

September 17, 2003

The NOR Gate

AY B

° This is a NOR gate. It is a combination of an OR gate followed by an inverter. It’s truth table shows this…

° NOR gates also have several

interesting properties…

• NOR(a,a)=(a+a)’ = a’ = NOT(a) • NOR’(a,b)=(a+b)’’ = a+b = OR(a,b) • NOR(a’,b’)=(a’+b’)’ = ab = AND(a,b)

ABY 001 010 100

110

ENGIN112 L7: More Logic Functions

September 17, 2003

Functionally Complete Gates

° Just like the NAND gate, the NOR gate is functionally complete…any logic function can be implemented using just NOR gates.

° Both NAND and NOR gates are very valuable as any design can be realized using either one.

° It is easier to build an IC chip using all NAND or NOR gates than to combine AND,OR, and NOT gates.

° NAND/NOR gates are typically faster at switching and cheaper to produce.

ENGIN112 L7: More Logic Functions

September 17, 2003

NOR Gates into Other Gates

(what are these circuits?)

AY

NOT Gate

A

B

Y

A

OR Gate

Y B

ENGIN112 L7: More Logic Functions

AND Gate

September 17, 2003

The XOR Gate (Exclusive-OR)

A B

° This is a XOR gate. ° XOR gates assert their output

when exactly one of the inputs is asserted, hence the name. ° The switching algebra symbol for this operation is , i.e. 1 1 = 0 and 1 0 = 1.

ENGIN112 L7: More Logic Functions

Y

ABY 000 011 101 110

September 17, 2003

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