Download presentation

Presentation is loading. Please wait.

Published byDuane Yeates Modified over 2 years ago

1
Boolean Logic ITI 1121 N. El Kadri

2
2 What is a switching network? Switching Network X1X1 XmXm X2X2 Z1Z1 ZmZm Z2Z2 Combinatorial Network: A stateless network. The output is completely determined by the values of the input. Sequential Network: The network stores an internal state. The output is determined by the input, and by the internal state.

3
3 Logic Functions: Boolean Algebra INVERTER XX’ If X=0 then X’=1 If X=1 then X’=0 OR ABAB C=A+B If A=1 OR B=1 then C=1 otherwise C=0 ABAB C=A·B If A=1 AND B=1 then C=1 otherwise C=0 AND

4
4 Boolean expressions and logic circuits Any Boolean expression can be implemented as a logic circuit. X = [A(C+D)]’+BE CDCD C+D [A(C+D)]’ [A(C+D)]’+BE BEBE BE A A(C+D)

5
5 Basic Theorems: Operations with 0 and 1 X+0 = X X0X0 C=X X+1 = 1 X1X1 C=1 X0X0 C=0 X·0 = 0 X1X1 C=X X·1 = X

6
6 Basic Theorems: Idempotent Laws X+X = X XXXX C=X XXXX X·X = X

7
7 Basic Theorems: Involution Law X (X’)’=X B C=X

8
8 Basic Theorems: Laws of Complementarity X+X’ = 1 X X’ C=1 X X’ C=0 X·X’ = 0

9
9 Expression Simplification using the Basic Theorems X can be an arbitrarily complex expression. Simplify the following boolean expressions as much as you can using the basic theorems. (AB’ + D)E + 1 = (AB’ + D)(AB’ + D)’ = (AB + CD) + (CD + A) + (AB + CD)’ = (AB’ + D)E + 1 = 1 (AB’ + D)(AB’ + D)’ = 0 (AB + CD) + (CD + A) + (AB + CD)’ = 1

10
10 Associative Law (X+Y)+Z = X+(Y+Z) XYXY Z C YZYZ X C

11
11 Associative Law (XY)Z = X(YZ) XYXY Z C YZYZ X C

12
12 First Distributive Law X(Y+Z) = XY+XZ

13
13 First Distributive Law X(Y+Z) = XY+XZ

14
14 First Distributive Law X(Y+Z) = XY+XZ

15
15 First Distributive Law X(Y+Z) = XY+XZ

16
16 First Distributive Law X(Y+Z) = XY+XZ

17
17 Second Distributive Law X+YZ = (X+Y)(X+Z)

18
18 Second Distributive Law X+YZ = (X+Y)(X+Z)

19
19 Second Distributive Law (A different proof) (X + Y)(X + Z)= X(X + Z) + Y(X + Z)(using the first distributive law) = XX + XZ + YX + YZ(using the first distributive law) = X + XZ + YX + YZ(using the idempotent law) = X·1 + XZ + YX + YZ(using the operation with 1 law) = X(1 + Z + Y) + YZ(using the first distributive law) = X·1 + YZ(using the operation with 1 law) = X + YZ(using the operation with 1 law)

20
20 Simplification Theorems (X + Y’)Y = XY XY + Y’Y = XY + 0 = XY XY’ + Y = X + Y (using the second distributive law) XY’ + Y = Y + XY’ = (Y + X)(Y + Y’) = (Y + X)·1 = X + Y XY + XY’ = X XY + XY’ = X(Y + Y’) = X·1 = X X + XY = X X(1 + Y) = X·1 = X (X + Y)(X + Y’) = X (X + Y)(X + Y’) = XX + XY’ + YX + YY’ = X + X(Y’ + Y) + 0 = X + X·1 = X X(X + Y) = X X(X + Y) = XX + XY = X·1 + XY = X(1 + Y) = X·1 = X

21
21 Examples Simplify the following expressions: W = [M + N’P + (R + ST)’][M + N’P + R + ST] W = M + N’P X = M + N’P Y = R + ST W = (X + Y’)(X + Y) W = XX + XY + Y’X + Y’Y W = X·1 + XY + XY’ + 0 W = X + X(Y + Y’) = X + X·1 = X

Similar presentations

Presentation is loading. Please wait....

OK

ALGEBRAIC EXPRESSIONS

ALGEBRAIC EXPRESSIONS

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on adaptive blind noise suppression Ppt on accounting standard 10 fixed assets Ppt on power situation in india Ppt on law against child marriage in yemen Ppt on effect of global warming on weather radio Ppt on pre ignition damage Ppt on business model canvas A ppt on sound Ppt on channel estimation Ppt on contract labour act 1970