1. Number Systems and Conversion

N = (a4a3a2a1a0a-1 a-2a-3)R = a4R4 + a3R3 + a2R2 + a1R1 + a0R0 + a-1R-1 + a-2R-2 + a-3R-3

Samples:

Decimal Conversion:

225.75 = 2*102 + 2*101 + 5*100 + 7*10-1 + 5*10-2

Binary to Decimal:

11100001.11 = 1*26 + 1*25 + 1*24 + 1*20 + 1*2-1 + 1*2-2

Hexadecimal to Decimal:

E1.C = 14*161 + 1*160 + 12*16-1

Decimal to Binary:

225.75 = ? in Binary
(Integral: 225) Method: Repeated "division" by 2
			Quotient: 0 1 3 7 14 28 56 112 225
			Remaindr: 1 1 1 0 0  0  0  1
(Fraction: .75) Method: Repeated "multiplication" by 2
			Fraction: .75  .5  0
			Integral:       1  1
Therefore: 225.7510 = 11100001.112

Decimal to Hexadecimal:

225.75 = ? in Hex
(Integral: 225) Method: Repeated "division" by 16
			Quotient:  0   14  225
			Remaindr:  14  1
(Fraction: .75) Method: Repeated "multiplication" by 16
			Fraction:  .75  0
			Integral:       12
Therefore: 225.7510 = E1.C16

2. Binary Codes:

  • For Digits 0 thru 9: 
    	Dec     8421      6311     2421      Excess-3 Gray     2-out-of-5
	-----------------------------------------------------------------
	0       0000	0000	0000	0011	0000	00011
	1       0001	0001	0001	0100	0001	00101
	2       0010	0011	0010	0101	0011	00110
	3       0011	0100	0011	0110	0010	01001
	4       0100	0101	0100	0111	0110	01010
	5       0101	0111	1011	1000	1110	01100
	6       0110	1000	1100	1001	1010	10001
	7       0111	1001	1101	1010	1011	10010
	8       1000	1011	1110	1011	1001	10100
	9       1001	1100	1111	1100	1000	11000
	------------------------------------------------------------------
  • Alphanumeric Codes: 
            ASCII Code
        char    Binary          Hex
        ---------------------------
        space   010 0000        20
        !       010 0001        21
        "       010 0010        22
        #       010 0011        23
        ...
        0       011 0000        30
        1       011 0001        31
        2       011 0010        32
        ...
        9       011 1001        39
        ...
        A       100 0000        40
        B       100 0001        41
        C       100 0010        42
        ...
        a       110 0000        60
        b       110 0001        61
        c       110 0010        62
        ...

3. Laws and Theorems:

Operations with 0 and 1:
(1) A + 0 = A
(2) A + 1 = 1 		.
(1d) A1 = A
(2d) A0 = 0
Idempotent Laws:
(3) A + A = A 		.
(3d) AA = A
Involution Law:
(4) (A')' = A 		.
.
Laws of Complementarity:
(5) A + A' = 1
.
(5d) AA' = 0
Commutative Laws:
(6) A + B = B + A
.
(6d) AB = BA
Associative Laws:
(7) (A + B) + C
= A + (B + C)
= A + B + C 		.
(7d) (AB)C
= A(BC)
= ABC
Distributive Laws:
(8) A(B + C)
= AB + AC 		.
(8d) A + BC
= (A + B)(A + C)
Simplification Theorems:
(9) AB + AB' = A
(10) A + AB = A
(11) (A + B')B = AB 		.
(9d) (A + B)(A + B')= A
(10d) A(A + B) = A
(11d) AB' + B = A + B
DeMorgan's Laws:.
(12) (A + B + C + *)'
= A'B'C'...
(13) [ f(A1,A2,...,AN, 0, 1, +, *) ]'
=f(A1',A2',...,AN',1, 0, *, +) 		.
(12d) (ABC...)'= A
= A' + B' + C' + ...
.
.
Duality:.
(14) (A + B + C + ...)^D
= ABC...
(15) [ f(A1,A2,..,AN,0,1,+,*) ]^D
= f(A1,A2,...,AN,1,0,*,+) 		.
14d) (ABC...)^D
= A + B + C + ...
.
.
Theorem for Multiplying Out and Factoring:.
(16) (A + B)(A' + C)
=AC + A'B
(15) [ f(A1,A2,..,AN,0,1,+,*) ]^D
.
(16d) AB + A'C
= (A + C)(A' + B)
Consensus Theorem:.
(17) AB + BC + A'C
= AB + A'C
.
(17d) (A + B)(B + C)(A' + C)
= (A + B)(A' + C)

4. For the following problem, find output Y and design a simpler network.

ANSWER:

Clearly, circuit output is
Y = A + B [ A B' + (A B + B ) ]

Therefore, Y = A + 0 + B = A + B

5. DeMorgan's Theorem

  1. [ SUM(Xn) ]' = PROD (Xn')
  2. [ PROD(Xn) ]' = SUM (Xn')

6. Another Example

Simplify: (A' + C' + D)(A + B' + C')(B + C')

ANSWER:

(A' + C' + D)(A + B' + C')(B + C') = (A' + C' + D)(AB + AC' + B'C' + BC' + C')
= (A' + C' +D)(AB + C') = A' + C' + BD

7. Consensus Theorem

Refer to Chapter 2. AB + A'C + BC = AB + A'C

8. An example:

Simplify F = ABC' + ABC'D' + ABD'E + A'EFG + CD'EG + CD'EG' (reduce to three terms)

ANSWER:

F = ABC' + ABD'E + A'EFG + CD'E = ABC' + CD'E + A'EFG

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