Activity 2.1.2
Have you ever wondered why we use the base-ten, or decimal, number system? Of course, we have ten fingers. The decimal number system that works so well for us is completely incompatible with digital electronics. Digital electronics only understand two states, ON and OFF. This is why digital electronics use the base-two, or binary, number system. In order for you to be able to design digital electronics, you will need to be proficient at converting numbers between the decimal and binary number systems.
In this activity you will learn how to convert numbers between the decimal and binary number systems.
Conclusion
1. The decimal number system has served humans well since the beginning of mankind. Ug the caveman didn’t call it the decimal number system, but he undoubtedly used his fingers to count objects in his world. If the decimal system is so good, why do computer and other digital electronic devices use the binary number system?
- every number have a value but its based on the other they come in
2. Now that we are using a number system other the decimal, it is important to properly subscript our numbers (i.e., 3510, 23410, 100102, etc.). Why is this so important? Provide at least three examples where neglecting to subscript numbers could lead to confusion.
- that way its not mistaken for 10.
3. Without performing the binary-to-decimal conversions, which of the following two binary numbers is the larger number :
Have you ever wondered why we use the base-ten, or decimal, number system? Of course, we have ten fingers. The decimal number system that works so well for us is completely incompatible with digital electronics. Digital electronics only understand two states, ON and OFF. This is why digital electronics use the base-two, or binary, number system. In order for you to be able to design digital electronics, you will need to be proficient at converting numbers between the decimal and binary number systems.
In this activity you will learn how to convert numbers between the decimal and binary number systems.
Conclusion
1. The decimal number system has served humans well since the beginning of mankind. Ug the caveman didn’t call it the decimal number system, but he undoubtedly used his fingers to count objects in his world. If the decimal system is so good, why do computer and other digital electronic devices use the binary number system?
- every number have a value but its based on the other they come in
2. Now that we are using a number system other the decimal, it is important to properly subscript our numbers (i.e., 3510, 23410, 100102, etc.). Why is this so important? Provide at least three examples where neglecting to subscript numbers could lead to confusion.
- that way its not mistaken for 10.
3. Without performing the binary-to-decimal conversions, which of the following two binary numbers is the larger number :
Activity 2.1.3
The first step in designing a new product is creating a design specifications document. These design specifications detail all of the features and limitations of the new product.
In digital electronics, the process of translating these design specifications into a functioning circuit starts with the creation of a truth table. A truth table is simply a list of all possible binary input combinations that could be applied to a circuit and the corresponding binary outputs that the circuit produces. Once the truth table is complete, a Boolean expression can easily be written directly from the truth table.
In this activity you will learn how to translate design specifications into truth tables and, in turn, write un-simplified logic expressions from these truth tables.
In future activities we will learn how to use Boolean algebra as well as a graphical technique called Karnaugh mapping to simplify these logic expressions.
Conclusion
1. A digital logic circuit with (2) inputs has (4) input combinations. One with (3) inputs has (8) combinations. One with (4) inputs has (16) combinations.
2. How many input combinations would a digital logic circuit have if it has (5) inputs? How about (6) input?
- (32) (64)
The first step in designing a new product is creating a design specifications document. These design specifications detail all of the features and limitations of the new product.
In digital electronics, the process of translating these design specifications into a functioning circuit starts with the creation of a truth table. A truth table is simply a list of all possible binary input combinations that could be applied to a circuit and the corresponding binary outputs that the circuit produces. Once the truth table is complete, a Boolean expression can easily be written directly from the truth table.
In this activity you will learn how to translate design specifications into truth tables and, in turn, write un-simplified logic expressions from these truth tables.
In future activities we will learn how to use Boolean algebra as well as a graphical technique called Karnaugh mapping to simplify these logic expressions.
Conclusion
1. A digital logic circuit with (2) inputs has (4) input combinations. One with (3) inputs has (8) combinations. One with (4) inputs has (16) combinations.
2. How many input combinations would a digital logic circuit have if it has (5) inputs? How about (6) input?
- (32) (64)
Activity 2.1.4
What does this circuit do? Does the circuit that I designed work? If you are able to analyze AOI logic circuits, you will be able to answer these questions. The first question frequently comes up when you need to determine the functionality of a previously designed circuit. The second question will always need to be answered whenever you design a new logic circuit.
When you analyze an AOI logic circuit, you can use one of two techniques. With the first technique, you determine the circuit’s truth table from which the output logic expression is derived. With the second technique, the order is reversed. The circuit’s logic expression is determined. The truth table is then derived using this expression.
Conclusion
1. In your own words, describe the process used to analyze a logic circuit where you first extract a truth table and then derive the logic expression.
- find your ones, then set up ur equations, then do the math.
2. Again, in your own words, describe the process used to analyze a logic circuit where you first extract the logic expression and then derive the truth table.
- find your ones, then set up ur equations, then do the math.
3. Did you find one of the processes easier than the other? Which one and why?
- easier at the end because it useful for saving space.
4. How can two logic equations be equal or equivalent?
- if the truth table comes out the same then all is good in the world.
What does this circuit do? Does the circuit that I designed work? If you are able to analyze AOI logic circuits, you will be able to answer these questions. The first question frequently comes up when you need to determine the functionality of a previously designed circuit. The second question will always need to be answered whenever you design a new logic circuit.
When you analyze an AOI logic circuit, you can use one of two techniques. With the first technique, you determine the circuit’s truth table from which the output logic expression is derived. With the second technique, the order is reversed. The circuit’s logic expression is determined. The truth table is then derived using this expression.
Conclusion
1. In your own words, describe the process used to analyze a logic circuit where you first extract a truth table and then derive the logic expression.
- find your ones, then set up ur equations, then do the math.
2. Again, in your own words, describe the process used to analyze a logic circuit where you first extract the logic expression and then derive the truth table.
- find your ones, then set up ur equations, then do the math.
3. Did you find one of the processes easier than the other? Which one and why?
- easier at the end because it useful for saving space.
4. How can two logic equations be equal or equivalent?
- if the truth table comes out the same then all is good in the world.
Activity 2.1.5
Would you pay $199 for a written specification for an MP3 player? Would you pay $299 for the schematics for a cell phone? Of course not. You don’t pay for the specifications or the schematics; you pay for the product itself.
You are not quite to the point where you can design an MP3 player or a cell phone, but you can design AOI logic circuits. In this activity you will learn how to implement AOI logic circuits from logic expressions. The logic expressions will be in either Sum-Of-Products (SOP) or Product-Of-Sums (POS) form.
Conclusion
1. Analyze each circuit to prove that they both produce the output Minterm=WXYZ.
- 1. is an and of ands, and 2. is an and of an and of an and.
2. First analyze the SOP version to determine the logic expression for F3 in SOP form. Use this expression to generate a truth table for the circuit.- in doing the work on another pieces of paper all worked out
3. How do the two truth tables compare? Is the column for F3 the same for both? They should be. If they are not the same, review your work and make any necessary corrections.
- compare the ones if there the same then its all good
Would you pay $199 for a written specification for an MP3 player? Would you pay $299 for the schematics for a cell phone? Of course not. You don’t pay for the specifications or the schematics; you pay for the product itself.
You are not quite to the point where you can design an MP3 player or a cell phone, but you can design AOI logic circuits. In this activity you will learn how to implement AOI logic circuits from logic expressions. The logic expressions will be in either Sum-Of-Products (SOP) or Product-Of-Sums (POS) form.
Conclusion
1. Analyze each circuit to prove that they both produce the output Minterm=WXYZ.
- 1. is an and of ands, and 2. is an and of an and of an and.
2. First analyze the SOP version to determine the logic expression for F3 in SOP form. Use this expression to generate a truth table for the circuit.- in doing the work on another pieces of paper all worked out
3. How do the two truth tables compare? Is the column for F3 the same for both? They should be. If they are not the same, review your work and make any necessary corrections.
- compare the ones if there the same then its all good
Activity 2.1.6
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?
1. Describe the process that you would use to simplify a logic expression using Boolean algebra.
- look at the rules and apply as needed
2. How do you know when you are finished simplifying and have arrived at the simplest equation?
- when you can't apply any more rules to it
3. Other than using Boolean algebra, how could you prove that two circuits are equivalent?
-truth table
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.
- the money for circuits are the same the money is saved in cost of wire
5. 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?
-Yes
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?
1. Describe the process that you would use to simplify a logic expression using Boolean algebra.
- look at the rules and apply as needed
2. How do you know when you are finished simplifying and have arrived at the simplest equation?
- when you can't apply any more rules to it
3. Other than using Boolean algebra, how could you prove that two circuits are equivalent?
-truth table
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.
- the money for circuits are the same the money is saved in cost of wire
5. 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?
-Yes
Activity 2.1.7
Despite all of the work done by George Boole, there was still more work to be done. Expanding on Boole’s studies, Augustus DeMorgan (1806-1871) developed two additional theorems that now bear his name. Without DeMorgan’s Theorems, the complete simplification of logic expression would not be possible.
1. How would you prove that the original Do-Nothing circuit and the simplified version are equivalent?
- because it dose nothing...
2. If each GATE cost 20¢ and you made 100,000 units, how much of the company’s money did you waste on your first project?
-6,000,000
Despite all of the work done by George Boole, there was still more work to be done. Expanding on Boole’s studies, Augustus DeMorgan (1806-1871) developed two additional theorems that now bear his name. Without DeMorgan’s Theorems, the complete simplification of logic expression would not be possible.
1. How would you prove that the original Do-Nothing circuit and the simplified version are equivalent?
- because it dose nothing...
2. If each GATE cost 20¢ and you made 100,000 units, how much of the company’s money did you waste on your first project?
-6,000,000