Boolean Algebra

Assalamualaikum. Hey there! Today in this entry, i will talk about Boolean Algebra. What is boolean? Boolean Algebra is actually founded by George Boole, an english mathematician in 1840s. So, enough with that historical story. COA is actually more harder and stressful to handle rather than history.

But if we study hard and we know the ’trick’, it will be quite easy. So, that is it, some introduction about boolean. J

In Boolean world, there are no such thing as ‘ ½’, ‘2/3’, ‘-1’ and so on. But it is come with True or False, or usually we use ‘0’and ‘1’. Let us start with Boolean identities first.

Another approach to express the logic function with logic equations:

• OR operator is written as +, as in A + B
• 0 +1 = 1 ------- 1 + 0 = 1

• AND operator is written as & , as in A & B
• 0 & 1 = 0 -------1 & 0 = 0

• NOT (inversion) operator is written as – or’  , as in A’ 
• 0’=1 ------ 1’= 0

Still get confused? Ok, i hope that u just do not feel boring.. 

Do not worry because..

Let me introduced you some video that might help you a lot about this topic. I have checked all of this video and it is easy to understand and fast to learn. :)

Actually, I want to make my own video about boolean, but my english is quite not good -_-. 

Boolean Algebra AND/OR/NOT

Boolean Algebra: Basic Law and Rules

Boolean Algebra: Basic Problem

Basic Problem Continued..

That is all for this post. Do not forget to comment. Bye!

by:

ABDUL WAFIY BIN SUZLY
B031210108
930408-01-5427

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Thanks for reading, please leave your comment here.

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K-Map

         Alright ladies and gents, today we’ll be learning about the Karnaugh Map otherwise known as the K-Map. The K-Map is an alternative method for simplifying a Boolean expression. It is suitable for inputs of 2 to 4 variables. This method is much simpler compared to using the Boolean algebra. There are 4 steps involved when making a karnaugh map which are:

1. By referring to the truth table, fill in all of the 1’s from the minterms in the truth table.

2. Group the 1’s following a simple set of rules. (We’ll get more in depth about those rules later).

3. List the groups, analyze it and simptuylify.

4. The simplified expressions are add together (*ORed).                  * Read as OR in past-tense form.

Let’s do some examples to help you get a clearer picture on what I’m rambling on about.

Example 1:  

Let’s say we are given 2 variables. Find the simplified Boolean expression.

Step 1: Make a truth table and a K-Map.

A	B	F	Minterms
0	0	0	A’B’
0	1	1	A’B
1	0	1	AB’
1	1	1	AB

B A	B’ (0)	B (1)
A’ (0)	O	1
A (1)	1	1

                                                                                               

* In the Karnaugh Map, each cells is filled with values that corresponds with the “F” column.

Step 2: Group the 1’s in the K-Map using a set of rules stated below.

• Each groups can consists of only 1’s.
• Only adjacent cells are allowed to be grouped.
• The group must be as large as possible meaning that it can have 2, 4, 8 or 16 of 1’s in it.
• The least number of grouhgvps formed the better.
• All 1’s must belong to a group even if it’s a group of one.
• Wrap around is allowed.
• Overlapping groups are allowed.

B     A	B’	B
A’	o	1
A	1	1

- The first group is B as variable B is the only

   variable that remains constant

B     A	B’	B
A’	o	1
A	1	1

    - The second group is A since A is the only variable that 

       remains constant.

Step 3: Skipping this step as we havjnkle already found the simplest Boolean expression.

Step 4:  Add together the groups.

                Answer = B + A or A + B

Example 2: Given 3 variables. Find the simplified Boolean expression.

Step 1:  Build a truth table and a K-Map.

A	B	C	F	Minterms
0	0	0	1	A’B’C’
0	0	1	1	A’B’C
0	1	0	1	A’BC’
0	1	1	1	A’BC
1	0	0	0	AB’C’
1	0	1	0	AB’C
1	1	0	1	ABC’
1	1	1	1	ABC

                             

BC A	B’C’	B’C	BC	BC’
A’	1	1	1	1
A	0	0	1	1

Step 2: Group the 1’s.

BC A	B’C’	B’C	BC	BC’
A’	1	1	1	1
A	0	0	1	1

                  - A’B’C’         - A’B’C

                 - A’BC            -A’BC’

BC A	B’C’	B’C	BC	BC’
A’	1	1	1	1
A	0	0	1	1

                 - A’BC            - A’BC’

                 - ABC             - ABC’

Step 3: Analyze.

For same letters in a column they will be brought down. For different letters in a column they will cancel each other out.

   Group 1:                                                                           Group 2:

   A’ B’ C’                                                                               - A’   B   C                                                                            

   A’ B’ C                                                                                - A    B    C           

   A’ B  C                                                                                 - A     B    C’                         

   A’ B  C’                                                                                - A’   B     C’   

   A’                                                                                                   B

                                                                                                               

Step 4: Add the groups together.

            

           Answer = A’ + B or B + A’

Below are some links that i found to be very helpful when learning about K-Maps:

http://www.wisc-online.com/Objects/ViewObject.aspx?ID=DIG5103

http://www.youtube.com/watch?v=-iF112AJGfc

http://www.laynetworks.com/Karnaugh%20Maps.htm

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