Activity 2.1.6
Introduction
Have you ever had an idea that you thought was so unique that when you told someone else about it, you simply could not believe they thought you were wasting your time? If so, you know how the mathematician George Boole felt in the 1800s when he designed a math system that, at the time, had no practical application. Today, however, his math system is the most important mathematical tool used in the design of digital logic circuits. Boole introduced the world to Boolean algebra when he published his work called “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities.”
In the same way that normal algebra has rules that allow you to simplify algebraic expressions, Boolean algebra has theorems and laws that allow you to simplify expressions used to create logic circuits.
By simplifying the logic expression, we can convert a logic circuit into a simpler version that performs the same function. The advantage of a simpler circuit is that it will contain fewer gates, will be easier to build, and will cost less to manufacture.
In this activity you will learn how to apply the theorems and laws of Boolean algebra to simplify logic expressions and digital logic circuits.
The moral of the story is to keep dreaming. Someday your grandchildren may be using something that you’re thinking about right now. When your grandparents were kids, do you think that they imagined someday that we would all have 10,000 songs in our pockets or a telephone in our backpacks?
Have you ever had an idea that you thought was so unique that when you told someone else about it, you simply could not believe they thought you were wasting your time? If so, you know how the mathematician George Boole felt in the 1800s when he designed a math system that, at the time, had no practical application. Today, however, his math system is the most important mathematical tool used in the design of digital logic circuits. Boole introduced the world to Boolean algebra when he published his work called “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities.”
In the same way that normal algebra has rules that allow you to simplify algebraic expressions, Boolean algebra has theorems and laws that allow you to simplify expressions used to create logic circuits.
By simplifying the logic expression, we can convert a logic circuit into a simpler version that performs the same function. The advantage of a simpler circuit is that it will contain fewer gates, will be easier to build, and will cost less to manufacture.
In this activity you will learn how to apply the theorems and laws of Boolean algebra to simplify logic expressions and digital logic circuits.
The moral of the story is to keep dreaming. Someday your grandchildren may be using something that you’re thinking about right now. When your grandparents were kids, do you think that they imagined someday that we would all have 10,000 songs in our pockets or a telephone in our backpacks?
Conclusion
1. Describe the process that you would use to simplify a logic expression using Boolean algebra.
I would use the Theorems to simply the equation. I’d probably have to start with the laws and demorgan’s theorems and then use the Boolean theorems.
2. How do you know when you are finished simplifying and have arrived at the simplest equation?
Then I can no longer apply anymore theorems to the equation.
3. Other than using Boolean algebra, how could you prove that two circuits are equivalent?
I could use the 3 laws to prove two circuits are equivalent.
4. If you worked for a company that manufactured the coffee vending machine that used the poorly designed circuit, how much money would your new design save the company annually if each GATE cost 15¢ and the company made 500,000 vending machines per year.
Each machine uses 6 gates so that’s 6x15¢=90¢. It costs 90¢ to make one, so times 500,000 that’s $450000.
As an experienced engineer, you earn $75 per hour. The total redesign took you two hours (including a coffee break). What would the company’s Return-On-Investment (ROI) be on your time?
ROI=(75,000/450,000)x100%=16.6%
Was it a good investment?
no
1. Describe the process that you would use to simplify a logic expression using Boolean algebra.
I would use the Theorems to simply the equation. I’d probably have to start with the laws and demorgan’s theorems and then use the Boolean theorems.
2. How do you know when you are finished simplifying and have arrived at the simplest equation?
Then I can no longer apply anymore theorems to the equation.
3. Other than using Boolean algebra, how could you prove that two circuits are equivalent?
I could use the 3 laws to prove two circuits are equivalent.
4. If you worked for a company that manufactured the coffee vending machine that used the poorly designed circuit, how much money would your new design save the company annually if each GATE cost 15¢ and the company made 500,000 vending machines per year.
Each machine uses 6 gates so that’s 6x15¢=90¢. It costs 90¢ to make one, so times 500,000 that’s $450000.
As an experienced engineer, you earn $75 per hour. The total redesign took you two hours (including a coffee break). What would the company’s Return-On-Investment (ROI) be on your time?
ROI=(75,000/450,000)x100%=16.6%
Was it a good investment?
no