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ICS 241

• Discrete Mathematics II– William Albritton, Information and Computer Sciences

Department at University of Hawai’i at Manoa

– For use with Kenneth H. Rosen’s Discrete Mathematics & Its Applications (5th Edition)

– Based on slides originally created by • Dr. Michael P. Frank, Department of Computer & Information

Science & Engineering at University of Florida

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Section 10.3 – Logic Gates

• Inverter, Or, And gate symbols.

• Multi-input gates.

• Logic circuits and examples.

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Logic Gate Symbols

• Inverter (logical NOT,Boolean complement).

• AND gate (Booleanproduct).

• OR gate (Boolean sum).

• XOR gate (exclusive-OR,sum mod 2).

x x

x

yx·y

x

yx+y

y

x x⊕y

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Multi-input AND, OR, XOR

• Can extend these gates to arbitrarilymany inputs.

• Two commonlyseen drawing styles:– Note that the second

style keeps the gate icon relatively small.

x1 x1x2x3x2x3

x1

⋮x5

x1…x5

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Class Exercise

• Exercise 3. (p. 718)– Each pair of students should use only one sheet

of paper while solving the class exercises

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NAND, NOR, XNOR

• Just like the earlier icons,but with a small circle onthe gate’s output.– Denotes that output is

complemented.

• The circles can also be placed on inputs.– Means, input is complemented

before being used.

x

y

x

y

y

x

xy

yx

yx

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Buffer

• What about an invertersymbol without a circle?

• This is called a buffer. It is the identity function.– It serves no logical purpose, but it represents an explicit

delay in the circuit.• This is sometimes useful for timing purposes.

• All gates, when physically implemented, incur a non-zero delay between when their inputs are seen and when their outputs are ready.

x x

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Combinational Logic Circuits

• Note: The correct word to use here is “combinational,” NOT “combinatorial!”– Many sloppy authors get this wrong.

• These are circuits composed of Boolean gates whose outputs depend only on their most recent inputs, not on earlier inputs.– Thus these circuits have no useful memory.

• Their state persists while the inputs are constant, but is irreversibly lost when the input signals change.

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Combinational Circuit Examples

• Examples– Light controlled by two switches– Half adder using OR/AND/NOT

• Full adder from half-adders

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Class Exercise

• Exercise 19. (p. 718)– Each pair of students should use only one sheet

of paper while solving the class exercises