Boolean Algebra: Minimum Sum-Of-Products Expression

In summary, the minimum Sum-Of-Product Expression for the given function is f = a(b+d)(b+c'+d'). This is one of two equivalent minsum forms and is the simplest expression that can be obtained for this function.
  • #1
Ithryndil
142
0

Homework Statement



Find the minimum Sum-Of-Product Expression for:
f = ab'c' + abd + ab'cd'

The Attempt at a Solution



By introducing the missing variable in term 1 and term 2 I can get an expression that has all the variables: a, b, c, and d.

I do so by:

f = ab'c'd + ab'c'd' + abcd + abc'd + ab'cd'

I can combine terms like so: (1 & 2),( 2 & 5), (3 & 4) I get:

f = ab'c' + ab'd' + abd

This hardly seems minimized from the original expression. Thanks for any help.
 
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  • #2
What you've stated is one of two equivalent minsum forms of that Boolean expression. There are several methods for arriving at these (consensus, Karaugh maps, Quine-McCluskey). I'd examine them for more info.

The original expression has a summand complexity (SC) of 3 and a literal complexity (LC) of 10. The minsum has an SC of 3 and an LC of 9 (as does the other). It isn't much simpler but it as simple as one can get.

--Elucidus
 
  • #3
Thanks, when you're learning about these concepts it is nice to have confirmation that you are doing things right. Normally it class we get the function down a term or two...or even to one term. So, when I got this down to three terms, with three variables in each term, it didn't really seem minimized. Thanks again!
 
  • #4
Karnaugh product of sums answer:

a(b+d)(b+c'+d')
 

What is Boolean Algebra?

Boolean Algebra is a branch of mathematics that deals with logical statements and operations, using only two values: true (represented by 1) and false (represented by 0). It is widely used in computer science and digital electronics to represent and manipulate logical states.

What is a Minimum Sum-Of-Products Expression?

A Minimum Sum-Of-Products (MSP) expression is a simplified form of a Boolean expression that represents the minimum number of ANDed terms (products) needed to achieve a specific logic output. It is also known as the canonical form and is useful for reducing the complexity of logical circuits.

How do you create a Minimum Sum-Of-Products Expression?

To create a Minimum Sum-Of-Products expression, you need to first write out the truth table for the given logic function. Then, identify the minterms (inputs that result in a true output) and group them into product terms with AND operations. Finally, combine all the product terms with OR operations to get the simplified expression.

What are the advantages of using Minimum Sum-Of-Products Expressions?

There are several advantages of using Minimum Sum-Of-Products expressions, including reduced complexity, improved efficiency, and easier implementation. By simplifying the expression, the number of logic gates and circuit complexity can be significantly reduced, resulting in faster and more efficient circuits.

How are Minimum Sum-Of-Products Expressions used in real-world applications?

Minimum Sum-Of-Products expressions are commonly used in digital electronics, particularly in the design of logical circuits and microprocessors. They are also used in software development for programming boolean logic and creating decision-making algorithms. Additionally, MSP expressions are used in database queries and search engines for filtering and retrieving data based on specific criteria.

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