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Nov 24, 2025

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Understanding Logic Circuits and Propositional Logic in Discrete Mathematics

E

eli gerszberg @eligerszberg_ebgh

Logic circuits and compound propositions are the building blocks of digital electronics and mathematical reasoning. These concepts help... Show more

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

Logic Circuits

Logic circuits are the physical implementations of logical operations. The NOT gate (also called an inverter) takes an input P and produces its opposite value (¬P). It essentially flips the truth value.

The OR gate combines two inputs (P and q) and produces an output that's true if at least one input is true. This is represented mathematically as P∨q.

The AND gate takes two inputs (P and q) and outputs true only if both inputs are true. This is written as P∧q in logical notation.

Quick Tip Remember that in digital circuits, "True" is represented by 1 and "False" by 0. This binary representation is the foundation of all computer operations!

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

Logic Circuit Examples

Logic circuits can implement complex logical expressions by combining different gates. For example, the expression (p ∨ ¬r) ∧ (¬p ∨ (q ∨ ¬r)) requires multiple gates connected in a specific way.

When drawing circuit diagrams, inputs flow from left to right through various gates. NOT gates (inverters) are used to create negations (¬p, ¬r), while AND and OR gates combine signals according to the logical expression.

The final output represents the truth value of the entire expression for any given input values of p, q, and r. This ability to model logical relationships is what makes digital computers possible!

Remember When constructing complex circuits, work step by step and follow the order of operations in the logical expression.

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

Compound Propositions

Logical statements come in three main types. A tautology is always true regardless of its variables' values (like p ∨ ¬p). A contradiction is always false (like p ∧ ¬p). A contingency depends on its variables, being true for some values and false for others.

An important relationship to know is that p → q (if p then q) is logically equivalent to ¬p ∨ q. This connection is useful for simplifying logical expressions.

When two propositions are logically equivalent, their biconditional (p ↔ q) is a tautology. We can write this equivalence as p ≡ q, meaning they have identical truth tables.

Study Hack When checking if two statements are logically equivalent, try constructing their truth tables. If the output columns match exactly, they're equivalent!

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

De Morgan's Laws

De Morgan's Laws are powerful tools for manipulating logical expressions. The key insight is that ¬(P∨q) = ¬P∧¬q and ¬(P∧q) = ¬P∨¬q. This means "not (P or q)" equals "not P and not q."

When applying De Morgan's Laws, first distribute the negation, then flip the operator (∨ becomes ∧, and vice versa). This technique helps simplify complex expressions.

Other important laws include identity laws P∧T=P,P∨F=PP∧T=P, P∨F=P, domination laws P∨T=T,P∧F=FP∨T=T, P∧F=F, and double negation ¬(¬P)=P¬(¬P)=P. These laws allow us to manipulate and simplify logical expressions systematically.

Pro Tip De Morgan's Laws are especially useful when designing circuits since they can help reduce the number of gates needed, making designs more efficient!

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

More Logical Laws

The commutative laws state that order doesn't matter in AND and OR operations P∨q=q∨P,P∧q=q∧PP∨q=q∨P, P∧q=q∧P. Similarly, associative laws tell us that grouping doesn't matter (P∨q)∨r=P∨(q∨r)(P∨q)∨r=P∨(q∨r).

Distributive laws allow us to expand expressions P∨(q∧r)=(P∨q)∧(P∨r). This works similar to distributing in algebra but with logical operators instead.

Absorption laws P∨(P∧q)=P,P∧(P∨q)=PP∨(P∧q)=P, P∧(P∨q)=P help eliminate redundancies in logical expressions. These laws, combined with those on the previous page, form a complete toolkit for manipulating logical expressions.

Application Note These logical laws aren't just theoretical—they're used daily in computer science to optimize programs, design efficient circuits, and verify that software works correctly!

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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

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Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

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I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

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Brad T

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Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

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In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

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I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

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Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

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THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

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This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Subjects

Arithmetic

Distributive Property

Equivalent Expressions

 

Arithmetic

•

50

•

Nov 24, 2025

•

5 pages

Understanding Logic Circuits and Propositional Logic in Discrete Mathematics

E

eli gerszberg

@eligerszberg_ebgh

Logic circuits and compound propositions are the building blocks of digital electronics and mathematical reasoning. These concepts help us understand how computers make decisions and how we can analyze logical statements systematically.

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

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Logic Circuits

Logic circuits are the physical implementations of logical operations. The NOT gate (also called an inverter) takes an input P and produces its opposite value (¬P). It essentially flips the truth value.

The OR gate combines two inputs (P and q) and produces an output that's true if at least one input is true. This is represented mathematically as P∨q.

The AND gate takes two inputs (P and q) and outputs true only if both inputs are true. This is written as P∧q in logical notation.

Quick Tip: Remember that in digital circuits, "True" is represented by 1 and "False" by 0. This binary representation is the foundation of all computer operations!

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

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Logic Circuit Examples

Logic circuits can implement complex logical expressions by combining different gates. For example, the expression (p ∨ ¬r) ∧ (¬p ∨ (q ∨ ¬r)) requires multiple gates connected in a specific way.

When drawing circuit diagrams, inputs flow from left to right through various gates. NOT gates (inverters) are used to create negations (¬p, ¬r), while AND and OR gates combine signals according to the logical expression.

The final output represents the truth value of the entire expression for any given input values of p, q, and r. This ability to model logical relationships is what makes digital computers possible!

Remember: When constructing complex circuits, work step by step and follow the order of operations in the logical expression.

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

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Compound Propositions

Logical statements come in three main types. A tautology is always true regardless of its variables' values (like p ∨ ¬p). A contradiction is always false (like p ∧ ¬p). A contingency depends on its variables, being true for some values and false for others.

An important relationship to know is that p → q (if p then q) is logically equivalent to ¬p ∨ q. This connection is useful for simplifying logical expressions.

When two propositions are logically equivalent, their biconditional (p ↔ q) is a tautology. We can write this equivalence as p ≡ q, meaning they have identical truth tables.

Study Hack: When checking if two statements are logically equivalent, try constructing their truth tables. If the output columns match exactly, they're equivalent!

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

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De Morgan's Laws

De Morgan's Laws are powerful tools for manipulating logical expressions. The key insight is that ¬(P∨q) = ¬P∧¬q and ¬(P∧q) = ¬P∨¬q. This means "not (P or q)" equals "not P and not q."

When applying De Morgan's Laws, first distribute the negation, then flip the operator (∨ becomes ∧, and vice versa). This technique helps simplify complex expressions.

Other important laws include identity laws P∧T=P,P∨F=PP∧T=P, P∨F=P, domination laws P∨T=T,P∧F=FP∨T=T, P∧F=F, and double negation ¬(¬P)=P¬(¬P)=P. These laws allow us to manipulate and simplify logical expressions systematically.

Pro Tip: De Morgan's Laws are especially useful when designing circuits since they can help reduce the number of gates needed, making designs more efficient!

# Logic Circuits P$\longrightarrow$ $\triangleright$$\longrightarrow$$\neg$P NOT gate (inverter) P$\longrightarrow$ $\longrightarrow$P$\v

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More Logical Laws

The commutative laws state that order doesn't matter in AND and OR operations P∨q=q∨P,P∧q=q∧PP∨q=q∨P, P∧q=q∧P. Similarly, associative laws tell us that grouping doesn't matter (P∨q)∨r=P∨(q∨r)(P∨q)∨r=P∨(q∨r).

Distributive laws allow us to expand expressions: P∨(q∧r)=(P∨q)∧(P∨r). This works similar to distributing in algebra but with logical operators instead.

Absorption laws P∨(P∧q)=P,P∧(P∨q)=PP∨(P∧q)=P, P∧(P∨q)=P help eliminate redundancies in logical expressions. These laws, combined with those on the previous page, form a complete toolkit for manipulating logical expressions.

Application Note: These logical laws aren't just theoretical—they're used daily in computer science to optimize programs, design efficient circuits, and verify that software works correctly!

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

4

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Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines

Mock Exam
Quiz
Flashcards
Essay

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user