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Topics for Monday, Jan. 25 Review homework DeMorgan’s Theorem relation to symbols Minterms and Maxterms prove x(x + y) = x 1. (10 points) Hint: Use duality and the proof already shown From Friday, x+xy = x  1 + xy = x(1+y) = x  1 = x Apply duality   , 1  0 x(x+y)

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Topics for monday jan 25 l.jpg
Topics for Monday, Jan. 25

  • Review homework

  • DeMorgan’s Theorem

    • relation to symbols

  • Minterms and Maxterms


Prove x x y x l.jpg
prove x(x + y) = x

  • 1. (10 points)Hint: Use duality and the proof already shown

  • From Friday, x+xy = x 1 + xy = x(1+y) = x 1 = x

  • Apply duality   , 1  0

    x(x+y)

    x(x+y)=(x+0)(x+y)

    = x+(0  y)

    = x+0 = x


Sum of products form l.jpg
Sum-of-Products form

2. (10 points) g = (a + c)(d + ef)

Let x=(a+c), y=d, z=ef and x(y+z) = xy+xz, so

(a+c)(d+ef) =(a+c)d + (a+c)ef

= ad + cd + aef + cef


Product of sums form l.jpg
Product-of-Sums form

3. (10 points) h=abc + d + e

Let (d+e)=x, ab = y, c = z and x+yz = (x+y)(x+z), so

abc + d + e = (ab + d + e)(c + d + e)

Let (d+e) = x and ab + d + e looks like x + yz, so

abc + d + e = (a + d + e)(b + d + e)(c + d + e)


Demorgan s theorem l.jpg
DeMorgan’s Theorem

  • DeMorgan’s Theorem provides the final link in taking any expression, including an expression with NAND or NOR, and converting it to S-o-P or P-o-S form.

  • (x+y)’ = x’ • y’ (x • y)’ = x’ + y’


Demorgan and symbols l.jpg
DeMorgan and Symbols

(x+y)’ = x’ • y’

f= (x+y)’= x  y

What symbol would express f = x’ • y’?

A logic mismatch between the symbol and the logic convention introduces the NOT operation


Nand nor equivalents l.jpg
NAND, NOR equivalents

(x+y)’ = x’ • y’

(x • y)’ = x’ + y’


Minterm l.jpg
Minterm

  • A minterm is composed of every input variable (or its complement) ANDed together

  • The truth table for a minterm in an n-variable system has one “1” and 2n - 1 ”0”’s

  • A minterm is denoted by a lowercase m and the (equivalent) number of the row on which the “1” occurs

    • a’b’c’ has a “1” for inputs 000 and is called minterm 0 and denoted m0

    • ab’c has a”1” for inputs 101 and is called minterm 5 and denoted m5


Truth tables to functions l.jpg
Truth Tables to Functions

  • Minterms provide a way to write the function for any truth table.

  • The minterms for every row where a “1” occurs are OR’ed together. ( x + 1 = 1)


Write a function l.jpg

a’b’c’

1

0

0

0

0

0

0

0

a’bc’

0

0

1

0

0

0

0

0

ab’c

0

0

0

0

0

1

0

0

abc’

0

0

0

0

0

0

1

0

a’b’c’

+ a’bc’

+ ab’c

+ abc’

Write a function

abc f

000 1

001 0

010 1

011 0

100 0

101 1

110 1

111 0

f =


Maxterm l.jpg
Maxterm

  • A Maxterm is composed of every input variable (or its complement) ORed together

  • The truth table for a Maxterm in an n-variable system has one “0” and 2n - 1 ”1”’s

  • A Maxterm is denoted by a uppercase M and the (equivalent) number of the row on which the “0” occurs

    • a’+b’+c’ has a “0” for inputs 111 and is called Maxterm 7 and denoted M7

    • a+b’+c has a”0” for inputs 010 and is called Maxterm 2 and denoted M2


Truth tables to functions12 l.jpg
Truth Tables to Functions

  • Maxterms provide a way to write the function for any truth table.

  • The Maxterms for every row where a “0” occurs are AND’ed together. ( x • 0 = 0)


Write a function maxterms l.jpg

(a+b+c’)

1

0

1

1

1

1

1

1

(a+b’+c’)

1

1

1

0

1

1

1

1

(a’+b+c)

1

1

1

1

0

1

1

1

(a’+b’+c’)

1

1

1

1

1

1

1

0

(a+b+c’)

(a+b’+c’)

(a’+b+c)

(a’+b’+c’)

Write a function (Maxterms)

abc f

000 1

001 0

010 1

011 0

100 0

101 1

110 1

111 0

f =


Canonical forms and lists l.jpg
Canonical Forms and Lists

  • The canonical minterm and Maxterm forms list every literal and operator

  • f = a’b’c’ + a’bc’ + ab’c + abc’

  • f= (a+b+c’)(a+b’+c’)(a’+b+c)(a’+b’+c’)

  • A list is a shorthand method of displaying which rows the 0’s and 1’s occupy

  • f = m(0,2,5,6) = M(1,3,4,7)


Homework for wednesday jan 27 l.jpg
Homework for Wednesday, Jan. 27

  • Problem 1. (10 points)

    • a) Write the electrical truth table for the gate symbol shown below

    • b) Write the operation for the gate symbol shown below when

      • Input=P.A. and Output=N.A

      • Input=P.A. and Output=P.A.

      • Input=N.A. and Output=N.A.

      • Input=N.A. and Output=P.A.


Problem 2 20 pts l.jpg

abcd f

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

1

0

1

0

111100000101

Problem 2. (20 pts)

a)Write the minterm expression for the function f(canonical form and list)

b) Write the Maxterm expression for the function f

(canonical form and list)


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