# How can I simplify boolean expressions using K-maps and truth tables?

• Live4eva_2
In summary, Warren managed to figure out how to reduce a function to a K-map. He had trouble with the ordering of the minterms and the values of the truth table, but he got help from a professor or TA and is now on the right track.
Live4eva_2
Hi...

I'm struggling in grasping how to reduce boolean expressions to K-maps/truth tables.
I drew a diagram in paint illustrating my steps in trying to reduce a function...
could someone please have a look and correct me in my method/point me in the right direction??

much appreciated...

Hope no-one minds but I posted this in the Comp_Sci section as well...because I know that this falls under both discrete maths and comp architecture...So rather check for my post there before wasting time providing help ,when someone else already has...

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• TRUTH.JPG
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Live4eva_2 said:
Hope no-one minds but I posted this in the Comp_Sci section as well...

That's a big no-no.

OK I marked the other thread as solved.
I managed to get a bit further with the K-maps.
Have a look at this .jpg.

I have managed to figure out the ordering of the minterms and the values of the truth table.The function in question is just an arbitrary one I made up so I'm not sure if that's a problem.

In my textbook the K-maps have curly braces indicating where (A,B,C are primed)...I don't quite understand that...

unless say F = A + (B'A') then A would appear primed and unprimed??

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• Kmap.jpg
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You're pretty much on the right track. In your first post you indicated the function was f = A + B'C, but in the second, it was f = A XOR B'C. I appears from your K-map that you actually intended the latter.

Also, in your K-map, you have incorrectly ordered the columns. They should read 00, 01, 11, 10. In this case you got lucky, because it doesn't matter.

Now you just need to circle the groups and write out the resulting expression.

- Warren

ah...this is the bit I don't understand.What is the method for writing out the expression once I have grouped everything??Could you explain?

Circle the largest groups you can, in powers of two. In other words, look for groups of 8, or 4, or 2, or 1 "ones" all adjacent to each other on the K-map.

Then, look at the row and column headings and figure out the term that represents each group. If you have a lone one that has a circle all by itself, it will result in a term with all of the inputs (A, B, and C in this case) represented. If you have a group of two, however, it'll depend on only two inputs -- one will be redundant.

It's really much easier shown on paper than described here in words. I suggest you see your professor or TA as soon as possible to get some one-on-one help.

- Warren

binary 100 -------This is my lone minterm. (ABC--->dependant on all inputs)

binary 110 ------
binary 111 ------These constitute my group of 2^1 minterms.(As you say,one input will be redundant.)

So...when all inputs are 1...I get a 1 for output.
Or...when C is 1...I get a 1 for output

Waitaminute,this is where that least significant bit comes in isn't it??Looking at the minterms I would say for the group of 2 minterms that C is redundant.Because of the fact that the truth table shows that if A and B are 1 the ouput is 1.So C is unnecessary??

But all this tells me is that I have ABC and AB appearing in my expression isn't it?

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## 1. What is a truth table and how is it used?

A truth table is a table used in logic and mathematics to determine the truth value of a logical expression. It lists all possible combinations of the input variables and shows the resulting outputs. It is used to help simplify and understand complex logical expressions.

## 2. What is a K-Map and how does it relate to a truth table?

A K-Map, also known as a Karnaugh map, is a graphical method used to simplify Boolean algebra expressions. It is based on the concept of grouping together adjacent squares in a table to identify patterns and simplify the expression. It is closely related to a truth table as both are used to simplify and analyze logical expressions.

## 3. How do you read a truth table?

A truth table is read from left to right, with the input variables listed in the order they appear in the expression. Each row represents a different combination of inputs, and the resulting output is listed in the last column. The output can be interpreted as either true or false, depending on the logical operators used in the expression.

## 4. What is the purpose of using a K-Map?

The purpose of using a K-Map is to simplify complex Boolean algebra expressions into a more manageable form. It allows for the identification of patterns and grouping of terms, making the expression easier to understand and potentially reducing the number of logic gates needed to implement the expression in a circuit.

## 5. How do you create a truth table and K-Map for a given logical expression?

To create a truth table, list all possible combinations of inputs for the expression and calculate the resulting output for each combination. To create a K-Map, draw a table with the input variables along the sides and group together adjacent squares based on the patterns in the expression. Both methods require a good understanding of logical operators and their truth tables.

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