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It has been my experience that students require much practice with circuit analysis to become proficient. To this end, instructors usually provide their students with lots of practice problems to work through, and provide answers for students to check their work against. While this approach makes students proficient in circuit theory, it fails to fully educate them.
Students don't just need mathematical practice. They also need real, hands-on practice building circuits and using test equipment. So, I suggest the following alternative approach: students should build their own "practice problems" with real components, and try to predict the various logic states. This way, the digital theory "comes alive," and students gain practical proficiency they wouldn't gain merely by solving Boolean equations or simplifying Karnaugh maps.
Another reason for following this method of practice is to teach students scientific method: the process of testing a hypothesis (in this case, logic state predictions) by performing a real experiment. Students will also develop real troubleshooting skills as they occasionally make circuit construction errors.
Spend a few moments of time with your class to review some of the "rules" for building circuits before they begin. Discuss these issues with your students in the same Socratic manner you would normally discuss the worksheet questions, rather than simply telling them what they should and should not do. I never cease to be amazed at how poorly students grasp instructions when presented in a typical lecture (instructor monologue) format!
I highly recommend CMOS logic circuitry for at-home experiments, where students may not have access to a 5-volt regulated power supply. Modern CMOS circuitry is far more rugged with regard to static discharge than the first CMOS circuits, so fears of students harming these devices by not having a "proper" laboratory set up at home are largely unfounded.
A note to those instructors who may complain about the "wasted" time required to have students build real circuits instead of just mathematically analyzing theoretical circuits:
                    What is the purpose of students taking your course?
If your students will be working with real circuits, then they should learn on real circuits whenever possible. If your goal is to educate theoretical physicists, then stick with abstract analysis, by all means! But most of us plan for our students to do something in the real world with the education we give them. The "wasted" time spent building real circuits will pay huge dividends when it comes time for them to apply their knowledge to practical problems.
Furthermore, having students build their own practice problems teaches them how to perform primary research, thus empowering them to continue their electrical/electronics education autonomously.
In most sciences, realistic experiments are much more difficult and expensive to set up than electrical circuits. Nuclear physics, biology, geology, and chemistry professors would just love to be able to have their students apply advanced mathematics to real experiments posing no safety hazard and costing less than a textbook. They can't, but you can. Exploit the convenience inherent to your science, and get those students of yours practicing their math on lots of real circuits!

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In order to familiarize students with the standard logic gate types, I like to given them practice with identification and truth tables each day. Students need to be able to recognize these logic gate types at a glance, or else they will have difficulty analyzing circuits that use them.

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In order to familiarize students with standard switch contact configurations, I like to given them practice with identification and truth tables each day. Students need to be able to recognize these ladder logic sub-circuits at a glance, or else they will have difficulty analyzing more complex relay circuits that use them.

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Some students with background in computers may ask if Boolean is the same as binary. The answer to this very good question is "no." Binary is simply a numeration system for expressing real numbers, while Boolean is a completely different number system (like integer numbers are to irrational numbers, for example). It is possible to count arbitrarily high in binary, but you can only count as high as "1" in Boolean.

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Boolean algebra is a strange math, indeed. However, once students understand the limited scope of Boolean quantities, the rationale for Boolean rules of arithmetic make sense. 1 + 1 must equal 1, because there is no such thing as "2" in the Boolean world, and the answer certainly can't be 0.

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Students need to be able to readily associate fundamental Boolean operations with logic circuits. If they can see the relationship between the ßtrange" rules of Boolean arithmetic and something they are already familiar with (i.e. truth tables), the association is made much easier.

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Students need to be able to readily associate fundamental Boolean operations with logic circuits. If they can see the relationship between the ßtrange" rules of Boolean arithmetic and something they are already familiar with (i.e. truth tables), the association is made much easier.

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Students need to be able to readily associate fundamental Boolean operations with logic circuits. If they can see the relationship between the ßtrange" rules of Boolean arithmetic and something they are already familiar with (i.e. truth tables), the association is made much easier.

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These three equivalencies will be vital for students to master as they study combinational logic circuits and complex relay logic circuits!

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In order to familiarize students with Boolean algebra and how it relates to logic gate circuitry, I like to give them daily practice with questions such as this. Students need to be able to recognize these logic gate types at a glance, and also be able to associate the proper Boolean expression with each one, or else they will have difficulty analyzing logic circuits later on.

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In order to familiarize students with Boolean algebra and how it relates to relay logic circuitry, I like to give them daily practice with questions such as this. Students need to be able to figure out how each one of these ladder logic circuits works, and also be able to associate the proper Boolean expression with each one, or else they will have difficulty analyzing more complex relay circuits later on.

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The process of converting gate circuits into Boolean expressions is really quite simple, if you proceed gate by gate. Have your students share whatever methods or "tricks" they use to write the expressions with the rest of the class.

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The process of converting gate circuits into Boolean expressions is really quite simple, if you proceed gate by gate. Have your students share whatever methods or "tricks" they use to write the expressions with the rest of the class.

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The process of converting gate circuits into Boolean expressions is really quite simple, if you proceed gate by gate. Have your students share whatever methods or "tricks" they use to write the expressions with the rest of the class.

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The process of converting relay logic circuits into Boolean expressions is not quite as simple as it is converting gate circuits into Boolean expressions, but it is manageable. Have your students share whatever methods or "tricks" they use to write the expressions with the rest of the class.

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The process of converting relay logic circuits into Boolean expressions is not quite as simple as it is converting gate circuits into Boolean expressions, but it is manageable. Have your students share whatever methods or "tricks" they use to write the expressions with the rest of the class.

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The process of converting relay logic circuits into Boolean expressions is not quite as simple as it is converting gate circuits into Boolean expressions, but it is manageable. Have your students share whatever methods or "tricks" they use to write the expressions with the rest of the class.

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This is not a very complicated function to express or implement, the point of this question being mostly to introduce students to a practical use of logic gates and Boolean algebra.

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The process of converting Boolean expressions into logic gate circuits is not quite as simple as converting gate circuits into Boolean expressions, but it is manageable. Have your students share whatever methods or "tricks" they use to write the expressions with the rest of the class.

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To be honest, I had fun writing the scenarios for different parts of this problem. The evolution of this alarm system is typical for an organization. Someone comes up with an idea, but it doesn't meet all the needs of someone else, so they input their own suggestions, and so on, and so on. Presenting scenarios such as this not only prepare students for the politics of real work, but also underscore the need to "what if?" thinking: to test the proposed solution before implementing it, so that unnecessary problems are avoided.

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First things first: did students remember to include the power supply connections to each IC? This is a very common mistake!
In order to successfully develop a solution to this problem, of course, students must research the "pinouts" of each integrated circuit. If most students simply present the answer shown to them in the worksheet, challenge them during discussion to present alternative solutions.
Also, ask them this question: ßhould we connect the unused inputs to either ground or V_CC, or is it permissible to leave the inputs floating?" Students should not just give an answer to this question, but be able to support their answer(s) with reasoning based on the construction of this type of logic circuit.

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A       B       Output       0       0       1       0       1       1       1       0       0       1       1       1

A       B       Output       0       0       0       0       1       1       1       0       1       1       1       1

Notes:
Ask your students to explain exactly how they figured out the Öutput" states to fill in the blanks in the truth tables, for the different input combinations. Ask them also to compare and contrast this process with that of figuring out the truth table for a given logic gate circuit.

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A       B       C       Output       0       0       0       1       0       0       1       1       0       1       0       1       0       1       1       1       1       0       0       1       1       0       1       1       1       1       0       0       1       1       1       1

A       B       C       Output       0       0       0       0       0       0       1       0       0       1       0       0       0       1       1       0       1       0       0       0       1       0       1       1       1       1       0       1       1       1       1       1

Notes:
Ask your students to explain exactly how they figured out the Öutput" states to fill in the blanks in the truth tables, for the different input combinations. Ask them also to compare and contrast this process with that of figuring out the truth table for a given logic gate circuit.
It is especially educational if you ask your students to suggest techniques for quickly determining truth table states, based on certain features of the Boolean expression. For instance, there is a way we can tell the first four Öutput" states in the truth table (reading top to bottom) will be 0 without having to plug values into the expression for B and C. Discuss with your students how we can look at the expression, seeing A as a multiplier for the sum within the parentheses, and immediately conclude that half of the truth table outputs will be 0.

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Notes:
The commutative, associative, and distributive laws of Boolean algebra are identical to the respective laws in real number algebra. These should not be difficult concepts for your students to understand. The real benefit of working through these examples is to associate gate and relay logic circuits with Boolean expressions, and to see that Boolean algebra is nothing more than a symbolic means of representing electrical discrete-state (on/off) circuits. In relating otherwise abstract mathematical concepts to something tangible, students build a much better comprehension of the concepts.

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Most of these Boolean rules are identical to their respective laws in real number algebra. These should not be difficult concepts for your students to understand. Some of them, however, are unique to Boolean algebra, having no analogue in real-number algebra. These unique rules cause students the most trouble!
An important benefit of working through these examples is to associate gate and relay logic circuits with Boolean expressions, and to see that Boolean algebra is nothing more than a symbolic means of representing electrical discrete-state (on/off) circuits. In relating otherwise abstract mathematical concepts to something tangible, students build a much better comprehension of the concepts.

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Quite frequently (and quite distressingly), I meet students who seem to have the most difficult time relating algebraic rules in their general form to specific instances of reduction. For example, a student who cannot tell that the rule A + AB = A applies to the expression QR + R, or worse yet B + AB. This skill requires time and hard work to master, because it is fundamentally a matter of abstraction: leaping from literal expressions to similar expressions, applying patterns from general rules to specific instances.
Questions such as this help students develop this abstraction ability. Let students explain how they "made the connection" between Boolean rules and the given reductions. Often, it helps to have a student explain the process to another student, because they are better able than you to put it into terms the struggling students can understand.

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Quite frequently (and quite distressingly), I meet students who seem to have the most difficult time relating algebraic rules in their general form to specific instances of reduction. For example, a student who cannot tell that the rule A + AB = A applies to the expression QR + R, or worse yet B + AB. This skill requires time and hard work to master, because it is fundamentally a matter of abstraction: leaping from literal expressions to similar expressions, applying patterns from general rules to specific instances.
Questions such as this help students develop this abstraction ability. Let students explain how they "made the connection" between Boolean rules and the given reductions. Often, it helps to have a student explain the process to another student, because they are better able than you to put it into terms the struggling students can understand.

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Notes:
An interesting way to sharpen students' understanding of algebraic techniques is to have them view someone else's incorrect work and find the error(s). Ultimately, algebraic reduction is really just an exercise in pattern recognition. Anything you can do to help your students recognize the correct patterns will help them become better at using algebra.

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For some reason, many of my students (who enter my course weak in algebra skills) generally seem to have a lot of trouble with factoring, be it Boolean algebra or regular algebra. This is unfortunate, as factoring is a powerful analytical tool. The "trick," if there is any such thing, is recognizing common variables in different product terms, and identifying which of them should be factored out to reduce the expression most efficiently.
Like all challenging things, factoring takes time and practice to learn. There are no shortcuts, really.

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Notes:
Quite frequently (and quite distressingly), I meet students who seem to have the most difficult time relating algebraic rules in their general form to specific instances of reduction. For example, a student who cannot tell that the rule A + AB = A applies to the expression QR + R, or worse yet B + AB. This skill requires time and hard work to master, because it is fundamentally a matter of abstraction: leaping from literal expressions to similar expressions, applying patterns from general rules to specific instances.
Questions such as this help students develop this abstraction ability. Let students explain how they "made the connection" between Boolean rules and the given reductions. Often, it helps to have a student explain the process to another student, because they are better able than you to put it into terms the struggling students can understand.

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�

FGH + G = G (Rule 1)

�

[`C] + CF = [`C] + F (Rule 2)

�

[`AB]C + A    (Neither rule applies)

�

RS +[`R] = [`R] + S (Rule 2)

�

[`AB] + ABC = [`AB] + C (Rule 2)

�

[`AB]C + C = C (Rule 1)

�

[`R]V[`W] +[`R] = [`R] (Rule 1)

�

[`X]  [`Y] Z +[`XY]    (Neither rule applies)

�

[`J]  [`K] L M +[`J]K    (Neither rule applies)

�

[`E]HF + F[`E] = [`E]F (Rule 1)

Notes:
Many students find the substitution of Boolean variables (going from the A's and B's of canonical rules to the different variables of real expressions where the rules are to be applied) very mysterious and difficult. Problems such as this give them practice learning to identify the rules' patterns despite similarities or differences in the actual variables (letters) used.

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Notes:
Have your students explain the entire process they used in answering this question: simplifying the expression using Boolean algebra techniques, and developing a gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Have your students explain the entire process they used in answering this question: simplifying the expression using Boolean algebra techniques, and developing a gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Have your students explain the entire process they used in answering this question: simplifying the expression using Boolean algebra techniques, and developing a gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Have your students explain the entire process they used in simplifying the gate circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Have your students explain the entire process they used in simplifying the gate circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Have your students explain the entire process they used in simplifying the gate circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Have your students explain the entire process they used in simplifying the relay circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new relay circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Have your students explain the entire process they used in simplifying the relay circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new relay circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Have your students explain the entire process they used in simplifying the relay circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new relay circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Have your students explain the entire process they used in simplifying the relay circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new relay circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Just a preview of DeMorgan's Theorem here!

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This question introduces DeMorgan's Theorem via a process of discovery. Students, seeing that these equivalent gates pairs have the same functionality, should be able to discern a general pattern (i.e. a rule) for breaking long bars in Boolean expressions.

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There are many suitable references for students to be able to learn DeMorgan's Theorem from. Let them do the research on their own! Your task is to clarify any misunderstandings after they've done their jobs.

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Have your students demonstrate exactly what they did (step by step) to simplify this expression, sharing their problem-solving strategies with the whole class.

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Have your students demonstrate exactly what they did (step by step) to simplify this expression, sharing their problem-solving strategies with the whole class.

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Have your students demonstrate exactly what they did (step by step) to simplify this expression, sharing their problem-solving strategies with the whole class.
Ask your students to determine what a non-variable solution means for a circuit such as this in a practical sense. What would they suspect if they tried to simplify a digital circuit and obtained this kind of result?

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Notes:
Have your students explain the entire process they used in simplifying the gate circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new gate circuit from the simplified Boolean expression. By having your students share their thought processes with the whole class, you will increase the level of learning on the parts of presenter and viewer alike. Students presenting their solutions will gain a better understanding of how it works because the act of presenting helps consolidate what they already know. Students viewing the presentation will get to see another person's technique (rather than just the instructor's), which will allow them to see examples of how to do these processes cast in slightly different terms.

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Notes:
Ask your students to explain what advantages there may be to using the simplified relay circuit rather than the original (more complex) relay circuit shown in the question. What significance does this lend to learning Boolean algebra?
This is what Boolean algebra is really for: reducing the complexity of logic circuits. It is far too easy for students to lose sight of this fact, learning all the abstract rules and laws of Boolean algebra. Remember, in teaching Boolean algebra, you are supposed to be preparing students to perform manipulations of electronic circuits, not just equations.

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Notes:
Ask your students to explain what advantages there may be to using the simplified gate circuit rather than the original (more complex) gate circuit shown in the question. What significance does this lend to learning Boolean algebra?
This is what Boolean algebra is really for: reducing the complexity of logic circuits. It is far too easy for students to lose sight of this fact, learning all the abstract rules and laws of Boolean algebra. Remember, in teaching Boolean algebra, you are supposed to be preparing students to perform manipulations of electronic circuits, not just equations.

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Notes:
An important aspect of long "bars" for students to recognize is that they function as grouping symbols. When applying DeMorgan's Theorem to breaking these bars, students often make the mistake of ignoring the grouping implicit in the original bars.
I highly recommend you take your class through the exercise suggested in the answer, for those who do not understand the nature of the mistake. Let students draw each expression's equivalent circuit on the board in front of the class so everyone can see, and then let them observe the dramatic change spoken of at the place where the mistake is made. If students understand what DeMorgan's Theorem means for an individual gate (Neg-AND to NOR, Neg-OR to NAND, etc.), the gate diagrams will clearly reveal to them that something has gone wrong at that step.
For comparison, perform the same step-by-step translation of the proper Boolean simplification into gate diagrams. The transitions between diagrams will make far more sense, and students should be able to get a "circuit's view" of why complementation bars function as grouping symbols.

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Notes:
The Boolean simplification for this particular problem is tricky. Remind students that complementation bars act as grouping symbols, and that parentheses should be used when in doubt to maintain grouping after "breaking bars" with DeMorgan's Theorem.
Ask your students to compare the ßimplified" circuit with the original circuit. Are any advantages apparent to the version given in the answer? Certainly, the Boolean expression for that version of the circuit is simpler compared to that of the original circuit, but is the circuit itself significantly improved?
This question underscores an important lesson about Boolean algebra and logic simplification in general: just because a mathematical expression is simpler does not necessarily mean that the expression's physical realization will be any simpler than the original!

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Notes:
Not only is the method shown in the answer not the only valid solution, but it may even be the worst one! Your students should be able to research or invent alternative inverter connections, so after asking them to present their alternatives, ask the class as a whole to decide which solution is better. Ask them to consider electrical parameters, such as propagation delay time and fan-out.

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Notes:
Not only is the method shown in the answer not the only valid solution, but it may even be the worst one! Your students should be able to research or invent alternative inverter connections, so after asking them to present their alternatives, ask the class as a whole to decide which solution is better. Ask them to consider electrical parameters, such as propagation delay time and fan-out.

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This question is a precursor to having students create combinational gate circuits using nothing but NAND or NOR gates.

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Notes:

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Gate universality is not just an esoteric property of logic gates. There are (or at least were) entire logic systems made up of nothing but one of these gate types! I once worked with a fellow who maintained gas turbine control systems for crude oil pumping stations. He told me that he has seen one manufacturer's turbine control system where the discrete logic was nothing but NAND gates, and another manufacturer's system where the logic was nothing but NOR gates. Needless to say, it was a bit of a challenge for him to transition between the two manufacturers' systems, since it was natural for him to "get used to" one of the gate types after doing troubleshooting work on either type of system.

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Notes:
An interesting feature of this circuit is the final three NAND gates: two NAND gates feeding into a third NAND gate is equivalent to two AND gates feeding into an OR gate, thanks to DeMorgan's Theorem!

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Notes:
This question is a really good one to ask your students how they arrived at a solution. It is easy enough to simply look at the given answer and repeat it, but of course the intent of this question is to get students to think how they might design such a circuit completely on their own.

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In my very first technical job, I worked as a CNC maintenance technician in a small machine shop, maintaining computer-controlled machine tools such as mills and lathes. A really neat project I got to work on at that job was the conversion of a 1970's era American-made machine tool to modern Japanese computer control. A lot of logic in that old machine tool was implemented using relays, and we replaced the cabinets full of relays with solid-state logic in the Japanese control computer. Actually, the solid state logic was a programmable logic controller or PLC function inside the Japanese control computer rather than discrete semiconductor logic gates. However, we very well could have replaced relays with hard-wired gates. The purpose of this question, if you haven't guessed by now, is to familiarize students with the concept of replacing electromechanical relays with semiconductor logic gates, especially identical logic gates such as NOR gates which are üniversal."

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Notes:
Here students see that even though two circuits are functionally identical (at least according to their respective Boolean expressions), they may not behave quite the same under adverse conditions (i.e. faulted switches or wiring). This is a very important thing for them to see, because it underscores the practical need to look beyond the immediate design criteria (Boolean function) and consider other parameters (failure mode).

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Notes:
Factoring really does seem to be a more difficult pattern-recognition skill to master than distribution, the latter being self-explanatory to many students. The purpose of this question is to get students to recognize and articulate the pattern-matching process involved with factoring. Once students have a working explanation of how to factor (especially if phrased in their own words), they will be better equipped to do so when needed.

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Notes:
This question stands as an example of how NAND gates may be used to construct different types of logic functions. In fact, with a sufficient quantity of NAND gates, any logic function may be built. This is why NAND gates are said to be üniversal."

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Notes:
This question stands as an example of how NOR gates may be used to construct different types of logic functions. In fact, with a sufficient quantity of NOR gates, any logic function may be built. This is why NOR gates are said to be üniversal."

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Notes:
This is a very practical application of DeMorgan's Theorem. Being able to use all NAND gates to implement an SOP function is a bonus over having to use separate AND and OR integrated circuit packages (one IC instead of two in this particular case).

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Notes:
This particular circuit is an example of how a combinational logic function may be implemented using nothing but NAND gates.

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Notes:
This particular circuit is an example of how a combinational logic function may be implemented using nothing but NAND gates.

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