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4.1 Switching Algebra

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1. Introduction

- Logic circuits are classified into two types:
- Combinational: whose outputs depend only on its current inputs.
- Sequential: depend not only on the current inputs but also on the past sequence of inputs, possibly arbitrarily far back in time.

- Analysis and Synthesis:
- Analysis start with a logic diagram and proceed to a formal description of the function performed by that circuit, such as a truth table or a logic expression.

A1

x=0 if x≠1

A1’

x=1 if x≠0

A2

if x=0, then x=1

A2’

if x=1, then x=0

A3

0·0=0

A3’

1+1=1

A4

1·1=1

A4’

0+0=0

A5

0·1=1·0=0

A5’

1+0=0+1=1

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- Synthesis do the reverse, starting with a formal description and proceeding to a logic diagram.

2. Axioms

Identities

T1

x+0=x

T1’

x·1=x

T2

x+1=1

T2’

x·0=0

Null elements

Idempotency

T3

x+x=x

T3’

x·x=x

Involution

T4

x =x

T4’

Complements

T5

x+x=1

T5’

x·x=0

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3. Theorems with One Variable

T6

x+y=y+x

Commutativity

T6’

x·y=y·x

T7

(x+y)+z=x+(y+z)

Associativity

T7’

(x ·y) ·z=x·(y·z)

T8

x·y+x·z=x·(y+z)

Distributivity

T8’

(x+y) ·(x+z)=x+y·z

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4. Theorems with multi-variable I

x+x·y=x

T9

Covering

T9’

x·(x+y)=x

T10

x·y+x·y=x

Combining

T10’

(x+y) ·(x+y)=x

T11

x·y+x·z+y·z=x·y+x ·z

Consensus

T11’

(x+y) ·(x+z) ·(y+z)= (x+y) ·(x+z)

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4. Theorems with multi-variable II

x+x+…+x=x

T12

T12’

x·x ·…· x=x

T13

x1 · x2 ·…· xn = x1+x2+…+xn

T13’

x1+x2+…+xn = x1 · x2 ·…· xn

T14

F(x1 , x2, … xn,+, ·)= F(x1 , x2, … xn, · ,+)

T15

F(x1 , x2, … xn)=x1 ·F(1, x2, … xn)+ x1 ·F(0, x2, … xn)

F(x1 , x2, … xn)=[x1+F(0, x2, … xn)]·[x1+F(1, x2, … xn)]

T15’

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4. Theorems with multi-variable III

Generalized idempotency

DeMorgan’s theorems

Generalized DeMorgan’s theorems

Shannon’s expansion theorems

x

x

y

Z=x·y

y

x

x

Z=x ·y

y

y

Z=x+y

- If F(w,x,y,z)=(w·x)+(x·y)+(w·(x+z ))
Then according to T14

F(w,x,y,z)=

Z=x+y

(w+x) ·(x+y) ·(w+(x·z))

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- Examples

- Equivalent circuits according to DeMorgan’s theorem T13

x1 · x2 ·…· xn = x1+x2+…+xn

x1+x2+…+xn = x1 · x2 ·…· xn

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5. Duality

- Principle of Duality: Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and · and + are swapped throughout.

- Examples

(Covering)

x+x·y=x

x·(x+y)=x

x·y+x·z+y·z=x·y+x ·z

(Consensus)

(x+y) ·(x+z) ·(y+z)= (x+y) ·(x+z)

(DeMorgan’s theorems)

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- Consider the following statement
x+x·y=x (T9)

x·x+y=x (According the principle of duality)

x+y=x (According theorem T3’)

How absurd it is! Where did we go wrong?

The problem is in operator precedence.

Actually x+x·y=x+(x·y) ∴ x·(x+y)=x

Operator precedence: ( ), AND, OR

- Sum-of-products expression: is a logic sum of product terms. (e.g. z+w·x·y+x·y·z)

- Product-of-sums expression : is a logic product of sum terms.
[ e.g. z·(w+x+y)·(x+y+z) ]

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6. Standard Representations of Logic

Functions

- Truth table: The brute-force representation simply lists the output of the circuit for every possible input combination.

Normal term: is a product or sum term in which no variable appears more than once.

e.g. z, w·x·y, x·y·z, w+x+y, x+y+z

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- Minterm: An n-variable minterm is a normal product term with n literals. There are 2n such product terms..

- Maxterm: An n-variable maxterm is a normal sum term with n literals. There are 2n such sum terms.

Row

x y z

F

minterm

maxterm

0

1

2

3

4

5

6

7

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

1

0

0

1

1

0

1

1

x·y·z

x·y·z

x·y·z

x·y·z

x·y·z

x·y·z

x·y·z

x·y·z

x+y+z

x+y+z

x+y+z

x+y+z

x+y+z

x+y+z

x+y+z

x+y+z

Canonical Sum

Canonical Product

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Minterm or maxterm number

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- Examples

- Write the canonical sum and product for each of the following logic function:

- Solution:

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- We have now learned five possible representations for a combinational logic function:

- A truth table

- An algebraic sum of minterms, the canonical sum.

- A minterm list using the ∑ notation.

- An algebraic product of maxterms, the canonical product.

- A maxterm list using the ∏ notation.