4 1 switching algebra
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Return. Next. 4.1 Switching Algebra. 1. Introduction. Logic circuits are classified into two types: Combinational: whose outputs depend only on its current inputs.

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4.1 Switching Algebra

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4 1 switching algebra

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4.1Switching Algebra

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.


4 1 switching algebra1

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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4.1Switching Algebra

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

2. Axioms


4 1 switching algebra2

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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4.1Switching Algebra

3. Theorems with One Variable


4 1 switching algebra3

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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4.1Switching Algebra

4. Theorems with multi-variable I


4 1 switching algebra4

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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4.1Switching Algebra

4. Theorems with multi-variable II


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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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4.1Switching Algebra

4. Theorems with multi-variable III

Generalized idempotency

DeMorgan’s theorems

Generalized DeMorgan’s theorems

Shannon’s expansion theorems


4 1 switching algebra6

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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4.1Switching Algebra

  • Examples

  • Equivalent circuits according to DeMorgan’s theorem T13


4 1 switching algebra7

x1 · x2 ·…· xn = x1+x2+…+xn

x1+x2+…+xn = x1 · x2 ·…· xn

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4.1Switching Algebra

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)


4 1 switching algebra8

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4.1Switching Algebra

  • 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


4 1 switching algebra9

  • 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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4.1Switching Algebra

6. Standard Representations of Logic

Functions

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


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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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4.1Switching Algebra

  • 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.


4 1 switching algebra11

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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4.1Switching Algebra

Minterm or maxterm number


4 1 switching algebra12

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4.1Switching Algebra

  • Examples

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

  • Solution:


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4.1Switching Algebra

  • 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.


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