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COS 250 Discrete Structures. Assoc. Prof. Svetla Boytcheva Spring semester 2011. Lecture № 10. Boolean Algebra. Boolean Functions Representing Boolean Functions Logic Gates Minimization of Circuits. Boolean Algebra.

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COS 250 Discrete Structures

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Cos 250 discrete structures

COS 250Discrete Structures

Assoc. Prof. Svetla Boytcheva

Spring semester 2011


Lecture 10

Lecture № 10

Boolean Algebra


Cos 250 discrete structures

  • Boolean Functions

  • Representing Boolean Functions

  • Logic Gates

  • Minimization of Circuits


Boolean algebra

Boolean Algebra

The basic rules for simplifying and combining logic gates are called Boolean algebra in honour of George Boole (1815 – 1864) who was a self-educated English mathematician who developed many of the key ideas.


Table 1 11 1

TABLE 1 (11.1)


Cos 250 discrete structures

NOT


Cos 250 discrete structures

AND


Cos 250 discrete structures

OR


Figure 1 11 1

FIGURE 1 (11.1)

FIGURE 1


Table 2 11 1

TABLE 2 (11.1)


Table 3 11 1

TABLE 3 (11.1)


Cos 250 discrete structures

x

f0

f1

f2

f3

0

1

0

0

0

1

1

0

1

1

n=1 - Boolean functions with 1 argument

f0(x)= const 0

f1 (x)=xidentity

f2 (x)= not xnegation (denotes like  xor)

f3 (x)= const 1


N 2 boolean functions with 2 arguments

x y

f0

f1

f2

f3

f4

f5

f6

f7

f8

f9

f10

f11

f12

f13

f14

f15

0 0

0 1

1 0

1 1

0

0

0

0

0

0

0

1

0

0

1

0

0

0

1

1

0

1

0

0

0

1

0

1

0

1

1

0

0

1

1

1

1

0

0

0

1

0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

1

1

0

1

1

1

1

0

1

1

1

1

n=2 - Boolean functions with 2 arguments

  • f0(x,y)=0;

  • f1(x,y)=x۸y - conjunction; (denotes like xy)

  • f3(x,y)=x;

  • f5(x,y)=y;


Cos 250 discrete structures

x y

f0

f1

f2

f3

f4

f5

f6

f7

f8

f9

f10

f11

f12

f13

f14

f15

0 0

0 1

1 0

1 1

0

0

0

0

0

0

0

1

0

0

1

0

0

0

1

1

0

1

0

0

0

1

0

1

0

1

1

0

0

1

1

1

1

0

0

0

1

0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

1

1

0

1

1

1

1

0

1

1

1

1

n=2

f6(x,y)=xy

f7(x,y)=x v y -

f8 (x,y)=x↓y-Pears

f9(x,y)= xy –equivalence (xy or xy )


Cos 250 discrete structures

x y

f0

f1

f2

f3

f4

f5

f6

f7

f8

f9

f10

f11

f12

f13

f14

f15

0 0

0 1

1 0

1 1

0

0

0

0

0

0

0

1

0

0

1

0

0

0

1

1

0

1

0

0

0

1

0

1

0

1

1

0

0

1

1

1

1

0

0

0

1

0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

1

1

0

1

1

1

1

0

1

1

1

1

n=2

  • f10(x,y)=not y;

  • f11(x,y)=y →x –reverse implication

  • f12(x,y)=not x

  • f13(x,y)=x →y –implication ‏


Cos 250 discrete structures

x y

f0

f1

f2

f3

f4

f5

f6

f7

f8

f9

f10

f11

f12

f13

f14

f15

0 0

0 1

1 0

1 1

0

0

0

0

0

0

0

1

0

0

1

0

0

0

1

1

0

1

0

0

0

1

0

1

0

1

1

0

0

1

1

1

1

0

0

0

1

0

0

1

1

0

1

0

1

0

1

1

1

1

0

0

1

1

0

1

1

1

1

0

1

1

1

1

n=2

f14(x,y)=x|y-Sheffer function

f15(x,y)=1;

f2 and f4not named - presents as


Sheffer stroke

Sheffer stroke

In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" or "↑", denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called the alternative denial, since it says in effect that at least one of its operands is false. In Boolean algebra and digital electronics it is known as the NAND operation ("not and").


Peirce arrow

Peirce arrow

Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician, and scientist, born in Cambridge, Massachusetts. In 1880 he discovered how Boolean algebra could be done via a repeated sufficient single binary operation (logical NOR)

The Peirce arrow,symbol for "(neither)...nor...", also called the Quine dagger.


Table 4 11 1

TABLE 4 (11.1)


Table 5 11 1

TABLE 5 (11.1)

Boolean functions with 1 argument


Table 6 11 1

TABLE 6 (11.1)


Table 1 11 2

TABLE 1 (11.2)


Table 2 11 2

TABLE 2 (11.2)


Figure 1 11 3

FIGURE 1 (11.3)

FIGURE 1 Basic Types of Gates.


Cos 250 discrete structures

XOR


Nand nor

NAND - NOR


And or and not represented by nand

AND, OR and NOT represented byNAND


Integral circuits

Integral circuits


Cos 250 discrete structures

Примери


Decoder

Decoder


Multiplexer

Multiplexer


Example

Example


Figure 2 11 3

FIGURE 2 (11.3)

FIGURE 2 Gates with n Inputs.


Figure 3 11 3

FIGURE 3 (11.3)

FIGURE 3 Two Ways to Draw the same Circuit.


Figure 4 11 3

FIGURE 4 (11.3)

FIGURE 4 Circuits that Produce the Outputs Specified in Example.


Figure 5 11 3

FIGURE 5 (11.3)

FIGURE 5 A Circuit for Majority Voting.


Table 1 11 3

TABLE 1 (11.3)


Figure 6 11 3

FIGURE 6 (11.3)

FIGURE 6 A Circuit for a Light Controlled by Two Switches.


Figure 7 11 3

FIGURE 7 (11.3)

FIGURE 7 A Circuit for a Fixture Controlled by Three Switches.


Table 2 11 3

TABLE 2 (11.3)

歐亞書局


Table 3 11 3

TABLE 3 (11.3)


Figure 8 11 3

FIGURE 8 (11.3)

FIGURE 8 The Half Adder.


Figure 9 11 3

歐亞書局

FIGURE 9 (11.3)

FIGURE 9 A Full Adder.

P. 765


Table 4 11 3

歐亞書局

TABLE 4 (11.3)

P. 765


Figure 10 11 3

歐亞書局

FIGURE 10 (11.3)

FIGURE 10 Adding Two Three-Bit Integers with Full and Half Adders.

P. 765


Figure 1 11 4

歐亞書局

FIGURE 1 (11.4)

FIGURE 1 Two Circuits with the Same Output.

P. 767


Useful links

Useful Links

http://www.chem.uoa.gr/applets/appletgates/appl_gates2.html

http://www.ee.surrey.ac.uk/Projects/Labview/Sequential/Course/01-Decoder/index.htm

http://www.allaboutcircuits.com/vol_4/chpt_9/2.html

http://www.electronics-tutorials.ws/combination/comb_1.html


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