# Elec 236 Logic Circuits - PowerPoint PPT Presentation

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Elec 236 Logic Circuits. Images from Chapter 3 Digital Systems 10 th Ed. by Tocci Prof. Tim Johnson. Basic Boolean Theorems. Communiative Theorems. x + y = y + x It doesn’t matter to the output where input x and input y are connected. Similarly… X*Y = Y*X

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Elec 236 Logic Circuits

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## Elec 236 Logic Circuits

Images from Chapter 3

Digital Systems 10th Ed. by Tocci

Prof. Tim Johnson

### Communiative Theorems

• x + y = y + x

• It doesn’t matter to the output where input x and input y are connected.

• Similarly…

• X*Y = Y*X

• The inputs to an AND gate don’t care which one is hook up where, switching the connections will not affect the output.

### Associative Theorems

• X + Y + Z = X + (Y + Z) = (X + Y) + Z

• XYZ = X(YZ) = (XY)Z

• In these cases, a three input OR/AND gate is the same as 2 two-input OR/AND gates where X and Y or Y and Z are connect to one of the input of the next OR/AND gate. The parenthesis represent the grouping of two of the inputs feeding into the next level.

### Distributive Theorems

• X(Y + Z) = XY +XZ

• (W + X)(Y + Z) = WY + WZ + XY + XZ

• In these cases, ordinary arithmetic rules apply and can alter the gate structure from 1) a two-input OR gate feeding an AND gate INTO two AND gates feeding a two-input OR gate, 2) two 2-input OR gates feeding a 2-input AND gate INTO four 2-input AND gates feeding a 3-input OR gate.

### Special rules for a 2-input OR gate where one input is an AND gate

• X + XY => X

• An input OR’ed with itself AND another input makes the other input unnecessary.

• X + XY => X + Y

• X + XY => X + Y

• If one of the OR inputs is the INVERSE of one of the AND inputs…you keep the single input by itself and drop the inverse input keeping the OR gate

### DeMorgan Theorems

• (X + Y) = X Y

• (X Y) = X + Y

• To implement these rules, looking at the left expression: break the bar and change the sign.

• This changes the output inversion to an inversion of the inputs and changes the type of gate used!

### Double Negative

• A double negative is an input, output, or group that has two bars across the input, output or group. Examples

• X, X+Y, ABC, L+MN

• Become

• X, X+Y, ABC, L+MN

• Ignore double bars of the same length over the same inputs. You can delete both bars

### Points to remember

• The letters used in these rules can represent groups:

• X ∙1 = X can also be written as AB∙1 = AB

• X + Y = Y + X ≈> DY + CE = CE + DY

• F + ABC = ABC + F

• C + LC = C + CL = CL + C

• X(Y + Z) = XY +XZ ≈> DMA + CMA = (D + C)MA

### More Points to Realize

• The Boolean rules apply across the equal sign meaning the change can go both ways:

• X(Y + Z) = XY + XZ means XY + XZ = X(Y + Z)

• Think of the equal sign as meaning 

CDE + ABC => CDE + CAB => C(DE + AB)

or skipping the second step you can just take the common term out of CDE + ABC as you would in Algebra.

### Moving on to more complex expressions

• BX + B is one of those 2-input OR gates using an AND gate as one of the inputs. Here’s why its correct: B(X+1) pulls the common term out. One (1) is a Boolean symbol that means always TRUE (or high). We have an elementary Boolean rule that deals with X+1 right?

X+1 = 1, so we can substitute in a 1 for the X+1 giving us: B·1. Don’t we have a rule for X·1 ? X AND 1 is X. Thus BX + B => B

### Writing a reduction solution to BX + B

BX + B

becomes

B(X + 1)

by distribution theorem

B∙1

by Basic Theorem #6

B

by Basic Theorem #2

### Inverted terms

• The common term can be a NOT input, X

• ABC + XYC => C(AB + XY)

• Carefully observe that the above expression does not meet all the criteria to apply the other rules for 2-input OR gates…dropping the other inputs to the AND gate. Those rules don’t apply here because the NOT input is not by itself (running solo).

### More expressions with NOT terms

• Take for example, X + XY = X + Y

BU + ACBU

rearranged:

BU + BUAC

This reduces to:

BU + AC

This does not work with BU +ABCU because

BU ≠ B∙U (DeMorgan rule applies here)