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L3 Logic Functions

# L3 Logic Functions - COSC 232 Digital Logic Logic Functions...

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ENGIN112 L7: More Logic Functions September 17, 2003 COSC 232 Digital Logic Logic Functions: NAND, NOR, XOR Prepared by: Russell Tessier Edited by: Michael Jack

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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) ° If there are N inputs, there are 2 2 N possible truth tables. For example, if 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. x 0 0 1 1 y 0 1 0 1 G 0 0 0 1

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ENGIN112 L7: More Logic Functions September 17, 2003 The NAND Gate ° This is a NAND gate. It is a combination of an AND gate followed by an inverter. ° 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) 0 1 1 1 0 1 1 1 0 1 0 0 Y B A A B Y
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 .

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ENGIN112 L7: More Logic Functions September 17, 2003 A Y A B Y Y A B NOT Gate AND Gate OR Gate NAND Gates into Other Gates (what are these circuits?)
ENGIN112 L7: More Logic Functions September 17, 2003 The NOR Gate ° This is a NOR gate. It is a combination of an OR gate followed by an inverter. ° 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) 0 1 1 0 0 1 0 1 0 1 0 0 Y B A A B Y

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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 (and cheaper) 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 NOT Gate OR Gate AND Gate NOR Gates into Other Gates (what are these circuits?) A Y Y A B A B Y

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ENGIN112 L7: More Logic Functions September 17, 2003 The XOR Gate (Exclusive-OR) ° This is a XOR gate.
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