Karnaugh Map by Eliza Joanne

The Karnaugh map or known as K-map is a simple method of minimizing Boolean expressions. K-map is a representation of the truth table in a grid-like manner.

Grouping 1’s in Karnaugh Map

• The groups only contain 1 s, no zeros allowed.
• There can only be adjacent grouping of cells only, no diagonal grouping.
• The number of groups can only be a power of 2(2ⁿ, where n = 0, 1, 2, 3, 4,…).
• The groups must be large as possible.
• All 1 must belong to a group.
• Overlapping of groups are allowed.
• Wrap around is allowed.
• There must be a few as possible groups.

In this case four variables are used, therefore the K-map has 16 outputs.

Sum-of-products (SOP)

Example 1: 

Let X = (A’B’C’D’) + (A’B’CD’) + (ABC’D’) + (ABC’D) + (AB’C’D’) + (AB’C’D)
     

                   
                  

Note: 
The labelling of 00, 01, 10, 11 is not in order. In the K-map 11 must be written before 10.

Following the rules as above:

Looking at the big box (4x4), AC’ does not change and from the wrap around A’B’D’ does not change. Hence, the simplification is AC’ + A’B’D’.

Example 2:

Let G = (A’B’C’D’) + (A’BC’D’) + (ABC’D’) + (AB’C’D’) + (AB’C’D)

                         

Following the rules as above:

Looking at the long box (1x4), C’D’ does not change and from the 2x2 box, AB’ does not change. Hence, the simplification is C’D’ + AB’C’.

 Product-of-sums (POS)

In a similar way as SOP, POS is simplified but using 0 instead of 1.

The rules of grouping 0 are the same as grouping 1. The changes are in grouping 0, no 1 can be included.

Example 1:

Let X = (A’B’C’D’) + (A’B’CD’) + (ABC’D’) + (ABC’D) + (AB’C’D’) + (AB’C’D)

Following the rules as above:

Looking at the 4x4 yellow box, AC does not change, from the green 4x4 box, A’D does not change and from the 1x4 box A’B’ does not change. Hence, the simplification is (A’+C’) x (A+D’) x (A+B).

Example 2:

Let H Let G = (A’B’C’D’) + (A’BC’D’) + (ABC’D’) + (AB’C’D’) + (AB’C’D)

                                                         

                                  

Following the rules as above:

Looking at the yellow 4x2 box, C does not change, from the green 4x4 box, BD does not change, and from the blue 4x4 box, A’D does not change. Hence, the simplification is C’ x (B’+D’) x (A+D’).

* With the known simplification the expression above can be found.

Boolean Algebra by AmiraHurriff

BOOLEAN ALGEBRA by AmiraHurriff
  Introduction :
A Boolean algebra is the combination of variables and operators.usually, it has one or more inputs and produces an output in the
Range of 0 and 1.  Laws Boolean Algebra expressions have been created to help reduce the number of logic gates(Boolean equation.
 These are consist of Boolean Algebra Laws:
•             commutative laws
•             Associative laws
•             Distributive laws
•             Identity laws
•             Zero and one laws
•             Inverse laws
•             DeMorgan’s laws

Types of law	AND form (gates)	OR form (gates)
Identity law	A . 1 = A	A + 0 = A
Zero and One law	A . 0 = 0	A + 1 = 1
Inverse law	A . A’ = 0	A + A’ = 1
Commutative law	A . B = B.A	A + B = B + A
Associative law	A.(B.C) = (A.B).C	A+(B+C)=(A + B)+C
Distributive law	A+(B.C)=(A+B).(A+C)	A.(B+C)=(A.B) + (A.C)
DeMorgan’s Law	(A’.B’)= A’ + B’	(A’ + B’)= A’.B’

 
De Morgan’s laws:
1.  Two separate terms NOR´ed together is the same as the two terms inverted (Complement) and AND´ed for example, A’+B’ = A’∙ B’.
2. Two separate terms NAND´ed together is the same as the two terms inverted (Complement) and OR´ed for example, A’∙B’ = A’ +B’.

Logic gate by sangkirthana

Logic gate

Logic gate

• A gate is digital circuit which either allow a signal pass through it or stop it.
• Its is commonly known as logic gate because it only allows the signal to pass through it only if certain logical condition is satisfied. 

Truth table and Boolean expression

• A table that shows all the possible combination of input and their corresponding output combination for logic gate is called a truth table or table of combination of that logic gate.
• George Boole device different kind of algebra called Boolean algebra deal wit logical statement  which having only 2 value namely either true or false value.
• In Boolean expression 1 denotes a truth statement and 0 denote the false statement
• The three basic operation in Boolean algebra are: 

1.  OR operation
2.  AND operation
3.  NOT operation

sum of product and product of sum by Shahariah Zulkifly

sum of product and product of sum

• logic function can be written in a canonical form
• there only 2 level of gates, AND and OR
• with a possible inversion on the final output
• sum of products represention of logical sum ( OR ) of product
• a product of sum is just there opposite
• E = ((A&B) + (A&C) + (B&C)) & (A&B&C)'
• E = (A&B&C') + (A&C&B') + (B&CA') (SOP)
• first equation has three levels of logic
• 2nd equation is in sum-of-products form

Integer Addition and Integer Subtraction by Shahariah Zulkifly

Integer Addition

*Remember this rules!!

·         0 + 0 = 0

·         0 + 1 = 1

·         1 + 0 = 1

·         1 + 1 = 1 0

 Example 1

·         5 + 7 = ?

Solution:

Step :

1.       Convert from decimal number to binary number

 Integer Subtraction

*Remember this rules!!

·         0 - 0 = 0

·         0 - 1 = 1 0 *borrow

·         1 - 0 = 1

·         1 - 1 = 1 

Example 1

·         1 0 1 – 0 1 1 = ?

Truth Tables by Mau Huey Ching

 

                                A  B  C D      } the four inputs

                                0  0   0  0                    

                                0  0   0  1                                         

                                0  0   1  0                            

                                0  0   1  1                           

                                0  1   0  0                         

                                0  1   0  1                       

                                0  1   1  0                         

                                0  1   1  1                        

                                1  0   0  0

                                1  0   0  1

                                1  0   1  0

                                1  0   1  1

                                1  1   0  0

                                1  1   0  1

                                1  1   1  0

                                 1  1   1  1

In column A consists of each a group of 0’s and 1’s

In column B consists of each two groups of 0’s and 1’s

In column C consists of each four groups of 0’s and 1’s

In column D consists of 0’s and 1’s alternately

Binary Multiplication and Binary Division by Mau Huey Ching

Binary Multiplication

Binary multiplication is an operation using basic multiplication consists 2 digits 0 and 1 only.

 

Binary Rules		Multiply
0 x 0 = 0		0
0 x 1 = 1		0
1 x 0 = 1		0
1  x  1 =  1	1

 

 

 

 

 

 

 

Example of Binary Multiplication

                            
                                                                            1010
                                                    x __1101_
                                                           1010
                                                         0000
                                                       1000
                                                   _1010____ 
                                                   _1111010_   

                                                           

Binary Division

Binary division also involves 2 digits 0 and 1 using basic division.

 Example of binary division:

                                
                                                         __111_
                                                   10    1110
                                                         _10_
                                                             11
                                                           _10_
                                                               10
                                                             _10_
                                                               00

 

 

 

One's complement and two's complement by Sangkirthana

One's complement

One's complement is operation to replace all the 0's to 1's and 1's to 0's.

    For example:                      

                                                (010010)₂

   The one’s complement =   (101101)₂

its simple as that..

Notes :

• For one's complement representation the left most bit is 0 for positive number and 1 for negative   number
• there also representation of both +0 and -0 which are 00000000 and 11111111 respectively.

Two's complement

Two's complement is form just like forming one's complement : complement all of the bit in number, then add 1,
For more understanding:
      Two’s complement = one’s complement + 1
                 

For example:                      

                                                (01101)₂

The two’s complement =       10010

                                                +        1

                                                  10011

Notes:

• For two's complement representation the left most bit is also 0 for positive and 1 for negative number, however this format does not have unfortune characteristic of signed magnitude and one's complement.

floating_point by amirahurriff

semua video ini berkaitan dengan floating point . bagaimana menggunakannya dan apabila ada operasi tambah ,tolak dan sbgainya.saya harap video2 ini dapat membantu anda untuk lebih memahami 'floating point' dgn lebih jelas. tajuk 'arithmetic in comp' ini tidak sesusah mana, jikalau anda fhm, dan banyak membuat latihan. :) .and btw thank you for watching these :) 

Number System Base and Number System Conversion by Eliza Joanne

Number System Base

Note: 
We do not need to memorize each binary digit for its equivalent decimal conversion as there is a clear pattern of repetition as shown above.

Decimal number

• Base of 10
• Composed of 10 digits (0 to 9)

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Binary number

• Base of 2
• Consist of only digits 0 and 1

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Hexadecimal number

• Base of 16
• Composed of alphanumeric values 0 to F
• Suitable to represent binary numbers with a multiple of 4 bits 

Number System Conversion

Decimal to binary

The conversion of decimal (base 10) to binary (base 2) is a simple division of the decimal number by two. The remainder from the division by 2 makes up a bit of the binary number.
Here are some examples (using the most common method) :

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For the decimal part .3125, we take the decimal and multiply by 2:

                                0.3125 x 2 = 0.625

Next we take .625 and do the same multiplication step:

                                0.625 x 2 = 1.25

The multiplication step is repeated by taking .025:

*Note: we only take the decimal part.

                                0.25 x 2 = 0.50

The step is again repeated until we obtain the value 1.00. We stop here because we have zero as the decimal part value.

Hence,                      0.50 x 2 = 1.00

The whole number part of the result is the binary digit.

Therefore, the binary value for 57.3125 is 110011.0101₂

Binary to decimal and hexadecimal to decimal

To convert binary number to decimal number, we add the product of the values and the digits of each bit. The values of each bit are 2 to the power of that bit.

Example:

Likewise for hexadecimal to decimal.
Example:

Binary to hexadecimal

To convert a binary number to hexadecimal, we divide it into groups of four bits starting from the right. If the number bits is not a multiple of four we add extra 0 bits to the left. This action is called padding.

Examples:

As there are 8 bits, we simply divide it to groups of four bits and find its equivalent hexadecimal value.








The number of bits above is not a multiple of 4, therefore we add two extra 0 bit to the left and then divide it to groups of 4 bits.

Hexadecimal to binary

This conversion is probably the easiest of all. We just take the hexadecimal value and substitute it with its corresponding binary digits.

For example:

4A₁₆ = 01001010₂

D64₁₆ = 110101100100₂

These topics may not be the heart of Computer Organization & Architecture but without this knowledge, understanding upcoming topics will be difficult. These are the basics in Computer Organization & Architecture.
Thank You.