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### Boolean Algebra – Part 2

### Canonical Forms

### Algebraic Simplification

### Incompletely Specified Functions

ECEn 224

Inversion

Inversion or Complement of a Function means

all 0 outputs become 1 and all 1 outputs become 0

ECEn 224

DeMorgan’s Laws

For complex expressions,

apply DeMorgan’s Laws successivelyDeMorgan’s Laws: (X+Y)’ = X’Y’ (XY)’ = X’ + Y’

Example:

F = A’B + AB’

F’ = (A’B + AB’)’

= (A’B)’•(AB’)’

= (A+B’)•(A’+B)

= AA’+AB+A’B’+BB’

= AB+A’B’

(X+Y)’ = X’Y’

(XY)’ = X’ + Y’

ECEn 224

DeMorgan’s Laws

Another Example:

F’ = [ ( a’b + 1)(cd + e’ + 0) ]’

= (a’b + 1)’ + (cd + e’ + 0)’

= [(a’b)’ • 1’] + (cd)’ • (e’)’ • 0’

= (a + b’) • 0 + (c’ + d’) • e • 1

Note: Expression is not simplified

ECEn 224

DeMorgan’s Laws - One Step Rule

[ f( X1, X2, … XN, 0, 1, +, •) ]’ = f( X1 ’, X2 ’, … XN ’, 1, 0, •, +)

1. Replace + with • and • with +

2. Replace 0 with 1 and 1 with 0

3. Replace all variables with the inverse

Be careful of hierarchy…

This is biggest source of errorsBefore beginning, surround all AND terms with parentheses

Like taking the dual

ECEn 224

DeMorgan’s Laws

Another Example:

F = ( a’b + 1)(cd + e’ + 0)

F = ( (a’ • b) + 1) • ( (c • d) + e’ + 0)

F’ = ( (a + b’) • 0) + ( (c’ + d’) • e • 1)

Variables and constants inverted!

Note: Expression is not simplified

ECEn 224

ECEn 224

Canonical Forms

- Is this equality true?
- AD + A’D’ + CD ?=? AD + A’D’ + A’C
- Hard to tell
- Both look minimized…
- But they look different…
- For many expressions there are multiple minimal forms
- Miminimizing is not a good way to determine equality

ECEn 224

Canonical Forms

- A canonical form is something written in the standard way
- Two expressions which are equal will have identical canonical forms
- Is there a standard (canonical) way of representing boolean expressions?
- Yes…

ECEn 224

Truth Tables Canonical?

f = AD + A’D’ + CD

f = AD + A’D’ + A’C

Same truth table, two different minimal expressions.

So, are truth tables canonical forms?

Yes, if we agree on a fixed column order.

ECEn 224

Minterm Expansion

f = AB + AC + B’C

- Each row in the truth table can be numbered
- Each row corresponds to an AND term in the SOP expression
- Call each row’s AND term a minterm(e.g., m0 = A’B’C’, m5 = AB’C)
- Combine minterms into SOP expressions
- Called minterm expansion or standard sum of products (standard SOP)

f = m1 + m5 + m6 + m7

f = m(1,5,6,7)

f = A’B’C + AB’C + ABC’ + ABC

ECEn 224

Minterms have names

- The order of the variables is significant!
- Given a minterm expression we can easily tell which minterm number it corresponds to

A’B’C’ 000 m0 AB’C’ 100 m4 AB’C 101 m5

- Given the minterm number we can derive the minterm expression.

m1 001 A’B’C m3 011 A’BC m6 110 ABC’

ECEn 224

Minterms

- Are these minterms of 4 variables?

abcd

ab’cd’

ab’c

a’bc’bd

Yes

Yes

no, only 3 variables

no, b more than once

ECEn 224

Minterm Expansion

- A minterm expansion is unique

f (A,B,C,D) =m(0,2,3,7)

- Useful for:
- Proving equality
- Shorthand for representing boolean expressions

ECEn 224

Maxterm Expansion

Any function can be written as a product of maxterms.

This is called the maxterm expansion or standard product of sums (standard POS)

f = AB + AC + B’C

Use the 0’s for the function to write the POS:

f(A,B,C) = M0 M2 M3 M4

f(A,B,C) = M(0, 2, 3, 4)

f = (A+B+C)(A+B’+C)(A+B’+C’)(A’+B+C)

ECEn 224

Maxterm Expansion

- Are these maxterms of 4 variables?

a + b + c + d

a + b’ + c + d’

a + b’ + d’

a’ + b + c’ + a’ + d

yes

yes

no, only 3 variables

no, a’ more than once

ECEn 224

Identifying Maxterms

- A little different from minterms
- Given a maxterm expression we can easily tell which maxterm number it corresponds to

A+B+C 000 M0 A’+B+C 100 M4 A’+B+C’ 101 M5

- Given the maxterm number we can derive the maxterm expression

M1 001 A+B+C’ M3 011 A+B’+C’ M6 110 A’+B’+C

ECEn 224

Maxterm Expansion

- Maxterm expansions are useful for the same things as minterm expansions
- Proving equality
- Shorthand for representing boolean expressions
- We will use the them when we want POS instead of SOP representations

ECEn 224

Minterm / Maxterm Relationships

For any function, if a minterm is in the Minterm Expansion,

the corresponding maxterm is not in the Maxterm Expansion.

f (A,B,C) = m1 + m5 + m6 + m7

f (A,B,C) = M0 M2 M3 M4

Makes it easy to convert between SOP and POS

ECEn 224

Minterm Example

- Is this equation true?

AD + A’D’ + CD = AD + A’D’ + A’C

- One method
- Draw truth table for each side
- Compare which minterms each side contains

Minterm expansions shorthand truth tables…

ECEn 224

Minterm Example

- Is this equation true?

AD + A’D’ + CD = AD + A’D’ + A’C

- Another method
- Expand each term into its minterms (X = XY’ + XY)
- Compare which minterms each side contains AD = AC’D + ACD AD = AC’D + ACD A’D’ = A’C’D’ + A’CD’ A’D’ = A’C’D’ + A’CD’ CD = A’CD + ACD A’C = A’CD’ + A’CD

ECEn 224

Minterm Example

Minimize the following:f(A,B,C) = m1 + m5 + m7

- Write out as SOPF = A’B’C + AB’C + ABC
- MinimizeF = B’C + AC

minterm expansions shorthand expressions…

ECEn 224

Maxterm Example

Minimize the following:f (A,B,C) = M1M5M7

- Write out as POSF = (A + B + C’)(A’ + B + C’)(A’ + B’ + C’)
- MinimizeF = (B + C’)(A’ + C’)

maxterm expansions shorthand expressions…

ECEn 224

Minterm/Maxterm Example

Convert the following function of A, B, C to POS:F = AB + C

- Write the minterm expansionF = m1 + m3 + m5 + m6 + m7(use truth table if it helps)
- Convert to maxterm expansionF = M0 • M2 • M4
- Write POS from maxterm expansionF = (A + B + C)(A + B’ + C)(A’ + B + C)
- Simplify if desiredF = (B + C)(A + C)

ECEn 224

ECEn 224

Algebraic SimplificationObjectives

- Why Simplify?
- Simpler equations less hardware to implement, faster, cheaper, less power

By inspection:f = A’B’C + AB’C + ABC’ + ABC

Simplifying:

f = A’B’C + AB’C + ABC’ + ABC

f = B’C + AB

4 3-input AND gates

1 4-input OR gate

XY’+XY=X

2 2-input AND gates

1 2-input OR gate

ECEn 224

Hardware Cost

- Exact cost depends on technology
- nMOS, CMOS, TTL, I2L, GaAs, FPGA, …
- In general:
- Fewer literals smaller/faster/lower power circuit
- Inverters don’t count
- Free in some technologies
- Insignificant difference in others

ECEn 224

Algebraic SimplificationWhich theorems to use?

- Suggestions:
- Know the identitiesand basic laws
- Memorize left side ofessential theorems
- Create duals on right as needed

ECEn 224

Four Methods of Algebraic Simplification

- Combine terms
- Eliminate terms
- Eliminate literals
- Add redundant terms

ECEn 224

Methods of Algebraic Simplification

- Combine terms

ABC’ + A’B’C + A’BC’

BC’ + A’B’C

- Eliminate terms
- Eliminate literals
- Add redundant terms

Use X Y + X Y’ = X

ECEn 224

Methods of Algebraic Simplification

- Combine terms
- Eliminate terms

A’B + B(A’D + C)

A’B + A’BD + BC

A’B + BC

- Eliminate literals
- Add redundant terms

Use X + X Y = X

ECEn 224

Methods of Algebraic Simplification

- Combine terms
- Eliminate terms
- Eliminate literals

A’B + A’B’C’D’ + ABCD’

A’(B + B’C’D’) + ABCD’

A’(B + C’D’) + ABCD’

A’B + A’C’D’ + ABCD’

A’C’D’ + B(A’ + ACD’)

A’C’D’ + B(A’ + CD’)

A’B + BCD’ + A’C’D’

- Add redundant terms

Use X + X’ Y = X + Y

ECEn 224

Methods of Algebraic Simplification

- Combine terms
- Eliminate terms
- Eliminate literals
- Add redundant terms

A’B’C + AB’C + ABC

A’B’C + AB’C + AB’C + ABC

B’C + AC

ECEn 224

Proving an Equation is True

- Construct a truth table for both sides
- Tedious, not very elegant
- Computers love this method
- Convert Both Sides to minterm/maxterm expansion
- Manipulate one side algebraically to equal the other
- Perform the same operation on both sides

ECEn 224

Perform the same operation on both sides

- Valid Operations
- Complement
- Construct the dual
- Invalid Operations
- Add (OR) a term to both sides (not reversible)
- Multiply (AND) both sides by a term (not reversible)
- Subtract a term from both sides (subtraction?)

ECEn 224

Proving an Equation Add a term to both sides

Prove Y = Z

Y = Z

Add 1 to both sides

Y + 1 = Z + 1

Use X + 1 = 1

1 = 1

Therefore Y = Z

NO! You cannot OR an arbitrary term with both sides.

However, you can duplicate an existing term:

Y = Z Y + Y = Z

ECEn 224

Proving an Equation Multiply (AND) both sides by a term

Prove Y = Z

Y = Z

Multiply both sides by 0

Y • 0 = Y • 0

Use X • 0 = 0

0 = 0

Therefore Y = Z

NO! You cannot AND an arbitrary term to both sides.

However, you can duplicate an existing term:

Y = Z Y • Y = Z

ECEn 224

Proving an Equation is False

- Show one combination of values for which the equation is false (counter example)
- Use Boolean algebra manipulation
- Reduce to sum of products form
- Minimize
- Compare the two sides
- Be careful! Some expressions have more than one minimal forms.

ECEn 224

ECEn 224

Incompletely Specified Functions

w

x

y

z

A

B

C

Known: N1 does not generate

outputs A B C with values

of 101 or 111

N1

N2

f

The X’s represent “don’t cares” – since those input combinations never occur, we don’t care what the output is.

ECEn 224

Incompletely Specified Functions

Multiple ways to choose those X’s when minimizing.

Cost of minimal solution will depend on values chosen.

f1 = A’B’C + AB’C’ + ABC’

f1 = A’B’C + AC’

f2 = A’B’C + AB’C’ + ABC’ + ABC

f2 = A’B’C + AC’(B’ + B) + AB(C’ + C)

f2 = A’B’C + AC’ + AB Worse than f1

ECEn 224

Incompletely Specified Functions

f1 = A’B’C + AC’

f3 = A’B’C + AB’C’ + AB’C + ABC’

f3 = A’C( B + B’) + AB’( C’ + C)

f3= A’C + AB’ Betterthanf1

f4 = A’B’C + AB’C’ + AB’C + ABC’ + ABC

f4 = (A’ + A)B’C+ AB’(C’ + ’C) + AB(C’ + C)

f4 = B’C+ AB’ + AB = B’C + A Best

ECEn 224

Incompletely Specified Functions

- If the function has n don’t cares
- Must solve 2n truth tables to use this method
- Ouch!
- A better way will be shown in a later chapter…

ECEn 224

Don’t Cares - Minterm Representation

Use d0 … d7 to represent don’t care minterms

F = m1 + m4 + m6 + d5 + d7F = m( 1, 4, 6) + d( 5, 7)

ECEn 224

Don’t Cares - Maxterm Representation

Use D0 … D7 to represent don’t care maxterms

F = M0 M2 M3 D5 D7

F = M(0, 2, 3) D(5, 7)

ECEn 224

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