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11/27/2025
Geometry
Logic and Reasoning: Conditional Statements
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Nov 27, 2025
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Understanding Conditional Statements in Logic and Reasoning
C
@cnnotes
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Logic and Reasoning: Conditional Statements
Conditional statements are the building blocks of logical reasoning. You've probably used them without even realizing it! When you think "If I finish my homework, then I can play video games," you're using a conditional statement.
These statements have two parts that work together to create logical connections. Understanding how they work will help you solve problems more efficiently and think more clearly.
Throughout this unit, we'll explore how to identify, create, and analyze different types of conditional statements.
💡 Quick Tip: Conditional statements are everywhere in real life - from game rules to scientific principles. Spotting them helps you become a better critical thinker!
Vocabulary of Conditional Statements
The language of logic has specific terms you need to know. A conditional statement is simply an if-then statement connecting two ideas.
Every conditional has two parts: the hypothesis (the "if" part) and the conclusion (the "then" part). For example, in "If it rains, then the ground gets wet," "it rains" is the hypothesis and "the ground gets wet" is the conclusion.
You can create related statements by rearranging or negating parts of a conditional. The converse swaps the hypothesis and conclusion. The inverse negates both parts. The contrapositive is the opposite of the converse.
The truth value tells us whether a statement is true or false, while negation simply means the opposite of a statement.
🔑 Remember: The way you arrange and negate parts of a conditional statement affects whether it remains true or becomes false!
Key Concept: Conditional Statements
Conditional statements follow a specific structure: "if p, then q" where p is the hypothesis and q is the conclusion. In mathematical notation, this is written as p → q.
Think of it this way: the hypothesis is the condition that must be met, and the conclusion is what happens as a result. For example, "If you heat water to 100°C, then it boils" shows a cause-and-effect relationship.
Venn diagrams offer a visual way to understand conditionals. The hypothesis set (p) is always inside the conclusion set (q). This means everything that satisfies the hypothesis must also satisfy the conclusion.
🧠 Visualization Tip: Think of the hypothesis as a smaller circle completely inside the larger conclusion circle. If something is in the hypothesis circle, it must also be in the conclusion circle!
Writing a Conditional Statement
Converting regular statements into conditional form is a skill you'll use often. The process is straightforward: identify what belongs in the hypothesis and conclusion, then add "if" and "then" in the right spots.
For example, take "A turtle is a reptile." The turtle is a specific animal (the smaller set), while reptiles are a larger group. So the hypothesis is "an animal is a turtle" and the conclusion is "it is a reptile."
When written properly: "If an animal is a turtle, then it is a reptile." This makes logical sense because all turtles are reptiles, but not all reptiles are turtles.
🔍 Practice Tip: When converting statements, look for which part is more specific (hypothesis) and which is more general (conclusion).
Related Conditional Statements
Each conditional statement has three related statements that provide different perspectives on the same information. Understanding all four types will strengthen your logical reasoning skills.
The original conditional (if p, then q) establishes a relationship. The converse (if q, then p) reverses this relationship and may not have the same truth value. For example, "If it's a bird, then it has wings" is true, but "If it has wings, then it's a bird" is false because bats have wings too.
The inverse (if not p, then not q) and contrapositive (if not q, then not p) involve negations. Interestingly, a conditional and its contrapositive always have the same truth value!
💡 Memory Hack: Think of the contrapositive as a "double flip" - you flip the order AND negate both parts of the original statement.
Writing Conditional Statements
Let's practice with a concrete example: "A square has four sides." When converted to a conditional statement, it becomes: "If an object is a square, then it has four sides." This statement is true because all squares have four sides.
The converse flips this around: "If an object has four sides, then it is a square." This is false because rectangles and rhombuses also have four sides but aren't squares.
The inverse states: "If an object is not a square, then it does not have four sides." This is false because rectangles aren't squares but still have four sides.
The contrapositive says: "If an object does not have four sides, then it is not a square." This is true - anything without exactly four sides cannot be a square.
⭐ Key Insight: Notice that the original conditional and its contrapositive are both true, while the converse and inverse are both false. This pattern often occurs in logical relationships!
More Examples of Conditional Statements
Sometimes all forms of a conditional statement can be true! Consider the example: "If you are studying math, then you are having fun!"
In this playful example, the converse ("If you are having fun, then you are studying math") is also true. Similarly, the inverse ("If you are not studying math, then you are not having fun") and contrapositive ("If you are not having fun, then you are not studying math") are both true.
This happens when there's a perfect one-to-one relationship between the hypothesis and conclusion. However, in most real-world situations, only the original conditional and its contrapositive share the same truth value.
🎬 Next Step: Watch the video in the lesson for more examples of conditional statements to deepen your understanding.
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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.
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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
Understanding Conditional Statements in Logic and Reasoning
C
@cnnotes
Logic and reasoning help us analyze statements and determine what's true or false. Conditional statements, also known as "if-then" statements, are a fundamental part of logical thinking that you'll use in math, science, and everyday decision-making.
Logic and Reasoning: Conditional Statements
Conditional statements are the building blocks of logical reasoning. You've probably used them without even realizing it! When you think "If I finish my homework, then I can play video games," you're using a conditional statement.
These statements have two parts that work together to create logical connections. Understanding how they work will help you solve problems more efficiently and think more clearly.
Throughout this unit, we'll explore how to identify, create, and analyze different types of conditional statements.
💡 Quick Tip: Conditional statements are everywhere in real life - from game rules to scientific principles. Spotting them helps you become a better critical thinker!
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Access to all documents
Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Vocabulary of Conditional Statements
The language of logic has specific terms you need to know. A conditional statement is simply an if-then statement connecting two ideas.
Every conditional has two parts: the hypothesis (the "if" part) and the conclusion (the "then" part). For example, in "If it rains, then the ground gets wet," "it rains" is the hypothesis and "the ground gets wet" is the conclusion.
You can create related statements by rearranging or negating parts of a conditional. The converse swaps the hypothesis and conclusion. The inverse negates both parts. The contrapositive is the opposite of the converse.
The truth value tells us whether a statement is true or false, while negation simply means the opposite of a statement.
🔑 Remember: The way you arrange and negate parts of a conditional statement affects whether it remains true or becomes false!
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Improve your grades
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Key Concept: Conditional Statements
Conditional statements follow a specific structure: "if p, then q" where p is the hypothesis and q is the conclusion. In mathematical notation, this is written as p → q.
Think of it this way: the hypothesis is the condition that must be met, and the conclusion is what happens as a result. For example, "If you heat water to 100°C, then it boils" shows a cause-and-effect relationship.
Venn diagrams offer a visual way to understand conditionals. The hypothesis set (p) is always inside the conclusion set (q). This means everything that satisfies the hypothesis must also satisfy the conclusion.
🧠 Visualization Tip: Think of the hypothesis as a smaller circle completely inside the larger conclusion circle. If something is in the hypothesis circle, it must also be in the conclusion circle!
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Improve your grades
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Writing a Conditional Statement
Converting regular statements into conditional form is a skill you'll use often. The process is straightforward: identify what belongs in the hypothesis and conclusion, then add "if" and "then" in the right spots.
For example, take "A turtle is a reptile." The turtle is a specific animal (the smaller set), while reptiles are a larger group. So the hypothesis is "an animal is a turtle" and the conclusion is "it is a reptile."
When written properly: "If an animal is a turtle, then it is a reptile." This makes logical sense because all turtles are reptiles, but not all reptiles are turtles.
🔍 Practice Tip: When converting statements, look for which part is more specific (hypothesis) and which is more general (conclusion).
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Improve your grades
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Related Conditional Statements
Each conditional statement has three related statements that provide different perspectives on the same information. Understanding all four types will strengthen your logical reasoning skills.
The original conditional (if p, then q) establishes a relationship. The converse (if q, then p) reverses this relationship and may not have the same truth value. For example, "If it's a bird, then it has wings" is true, but "If it has wings, then it's a bird" is false because bats have wings too.
The inverse (if not p, then not q) and contrapositive (if not q, then not p) involve negations. Interestingly, a conditional and its contrapositive always have the same truth value!
💡 Memory Hack: Think of the contrapositive as a "double flip" - you flip the order AND negate both parts of the original statement.
Sign up to see the contentIt's free!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Writing Conditional Statements
Let's practice with a concrete example: "A square has four sides." When converted to a conditional statement, it becomes: "If an object is a square, then it has four sides." This statement is true because all squares have four sides.
The converse flips this around: "If an object has four sides, then it is a square." This is false because rectangles and rhombuses also have four sides but aren't squares.
The inverse states: "If an object is not a square, then it does not have four sides." This is false because rectangles aren't squares but still have four sides.
The contrapositive says: "If an object does not have four sides, then it is not a square." This is true - anything without exactly four sides cannot be a square.
⭐ Key Insight: Notice that the original conditional and its contrapositive are both true, while the converse and inverse are both false. This pattern often occurs in logical relationships!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
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More Examples of Conditional Statements
Sometimes all forms of a conditional statement can be true! Consider the example: "If you are studying math, then you are having fun!"
In this playful example, the converse ("If you are having fun, then you are studying math") is also true. Similarly, the inverse ("If you are not studying math, then you are not having fun") and contrapositive ("If you are not having fun, then you are not studying math") are both true.
This happens when there's a perfect one-to-one relationship between the hypothesis and conclusion. However, in most real-world situations, only the original conditional and its contrapositive share the same truth value.
🎬 Next Step: Watch the video in the lesson for more examples of conditional statements to deepen your understanding.
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.
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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