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Course: Algebra (all content)>Unit 18

Lesson 9: Deductive and inductive reasoning

Using deductive reasoning

Sal uses deductive reasoning to prove an algebraic identity. Created by Sal Khan and Monterey Institute for Technology and Education.

Want to join the conversation?

• I got 0=0. Am I doing something wrong here???
(13 votes)
• No. You've just proved the theorem as 0=0 is true.
(37 votes)
• So I was watching these videos to try and gain a fuller understanding of these topics so I could do my homework better. My main struggle was understanding the Law of Syllogism and the Law of Detachment. However, I didn't hear Khan mention these two laws even once during any of these videos. Does anyone happen to know what these two are? If so, would you mind explaining it to me? Or if you heard it during the video at some point, could you tell me where he starts talking about them? If no one else has seen these two laws mentioned, I would like to know why Khan doesn't mention them in here. I can somewhat understand the concepts, but not how to apply it to actual problems. Thank you.
(10 votes)
• You generally will apply these concepts in algebra and geometry. Here's a few examples. The Law of Syllogism states that if we have the statements, "If p, then q" and, "If q, then r", then the statement, "If p, then r" is true. A nice way to conceptualize this is if a = 5, and 5 = b, then a = b. You will use this a lot in traditional geometry proofs. For example, if angle A is congruent to angle B because of the vertical angles theorem and if angle B is congruent to angle C because of the alternate interior angles theorem, then angle A is congruent to angle C because of the Law of Syllogism. You've just proved the corresponding angles theorem. The Law of Detachment states that if we have the statements "If p, then q" and "p" then the statement "q" is true. Another example (picking up from the end of the end of the last example) is, if angle A = 70 degrees, then angle C = 70 degrees because of the definition of congruent and if the measure of angle A = 70 degrees because it is given, then angle C = 70 degrees because of the Law of Detachment. Usually the Law of Detachment is pretty obvious and not directly stated.
(10 votes)
• How do you provide the reasoning for each step?
(0 votes)
• One way to do this is to write the property or definition that justifies each step. Sal talked through the reasoning, but if you have to provide them in written form, put them out to the right side of each step.

Justify (x + y)² = x² + 2xy + y²
(x + y)² = (x + y)(x + y) `definition of exponent a² = a∙a`
(x + y)(x + y) = (x + y)x + (x + y)y `definition of distribution a(b + c) = ab + ac`
(x + y)x + (x + y)y = x(x + y) + y(x + y) `commutative property of multiplication`
x(x + y) + y(x + y) = xx + xy + yx + yy `definition of distribution a(b + c) = ab + ac`
xx + xy + yx + yy = x² + xy + yx + y² `definition of exponent a∙a = a² `
x² + xy + yx + y² = x² + xy + xy + y² `commutative property of multiplication ab = ba`
x² + xy + xy + y² = x² + 2xy + y² `math fact 1 + 1 = 2`
Therefore, we have shown:
(x + y)² = x² + 2xy + y²
using deductive (logical) reasoning, divided into very small logical steps with all the reasoning (justifications) shown.
(9 votes)
• Would it be much easier to use the First Outer Inner Last (FOIL) method once you get to this step:
(x+y) (x+y)
First: x times x = x squared
Outer: x times y = xy
Inner: y times x = yx
Last: y times y = y squared
You would get the exact same thing its just easier to find
(1 vote)
• FOIL is a guideline for beginners who do not yet grasp the distributive property.
All steps in the video are mathematically sound reasoning.
While FOIL gives you the correct answer, it shortcuts the mathematical reasoning - the video is, in a way, proof why FOIL works since you can rearrange, using the commutative and associative properties, the intermediate result Sal gets with what FOIL produces.
You may want to investigate mathematical soundness and validity.

Keep Studying and
Keep Asking Questions!
(5 votes)
• So, to sum it all up, can we say that the mathematical processes including only logical operations come under deductive reasoning and the ones including assumptions come under inductive reasoning?
(2 votes)
• Industive reasoning uses reason, and patterns to come to a conclusion about something, while deductive reasoning uses facts, logic, and definitions to come to a conclusion about something.
(1 vote)
• What type of reasoning would you use to prove a conjecture?
(2 votes)
• Deductive reasoning. A conjecture is something you would form using inductive reasoning. Also, if you prove a conjecture using deductive reasoning, then it becomes a theorem.
(1 vote)
• Okay so Deductive reasoning starts with a fact, while Inductive reasoning is basically using patterns to make a probable solution. am I right?
(2 votes)
• What if the problem is a statement? How do you determine whether its deductive or inductive ?
(2 votes)
• Deduction is drawing a conclusion from something known or assumed. This is the type of reasoning we use in almost every step in a mathematical argument.
Mathematical induction is a particular type of mathematical argument. It is most often used to prove general statements about the positive integers.
So if the problem statement reads, prove that this equals that, it is deductive.
If the problem statement reads, prove that this equals that for all values of n, it is inductive.
(1 vote)
• Can someone answer this question?

If 2x-5=11, then x = 8.
Determine whether the converse is true or false. If true, create biconditional. If false, create a counterexample.

Converse: If x=8, then 2x-5=11.
My teacher said the converse is false because x=8 can be the solution to numerous equations, but if you substitute x=8 to 2x-5, doesn't it equal 11. So, shouldn't the converse be true.
(1 vote)
• If the hypothesis of a conditional is false, can you use the Law of Detachment/Syllogism?
(1 vote)

Video transcript

Use deductive reasoning and the distributive property to justify x plus y squared is equal to x squared plus 2xy plus y squared. Provide the reasoning for each step. Now when they say use deductive reasoning and all this stuff, it might seem like something daunting and new, but this is no different than what we've done in the past. In fact, we've done this very exact problem. So let's just do it step by step, show our logic, and that's essentially deductive reasoning. We're starting with a statement and we're going to deduce that this is going to be equal to something else. So let's start with x plus y squared. We know an exponent means to multiply something by itself that many times. So we know that this is the same thing, or we can deduce that it is the same thing, as x plus y times-- and I'll do this next x plus y in a different color-- times x plus y. That's what x plus y squared is. Now they say to use the distributive property. And this is a bit of review here. And we've seen it many, many, many times before. The distributive property just tells us that if we have a times b plus c-- I want to do as many colors as possible-- that this is equal to a times each of these terms. a times b plus a times c. It's called distributive property cause you're distributing the a in all of the terms in the expression that your multiplying a by. Now we can do the exact same thing here. Instead of an a, you could imagine this is an x plus y. And we can take this entire x plus y and we can distribute it on to both of the terms on this expression that it is multiplying. If this was an a, it'd be ax plus ay. Now that it's an x plus y, we multiply the x plus y times each of those terms. And that's just by the distributive property. So by the distributive property that's going to be equal to-- we'll distribute this on to each of them-- x plus y times x. And actually I don't even have to write the x after it. I could just write it there. It doesn't matter whether you multiply x times x plus y, or x plus y times x. Order doesn't matter. So that's that times that. And then it's going to be plus y times x plus y. And now we can apply the distributive property again. We have x being multiplied by x plus y, then we have a y being multiplied by x plus y. So let's just do that again. So then we get this is equal to x times x plus x times y. I'm going through great pains to keep the colors consistent. Plus y times x. Plus y times this x over here. Plus y times that y over there. I'm doing this a lot slower, and I'm not skipping any steps here. Now what do each of these things equal? x times x. That is the same thing as x squared. So this is equal to x squared. This right here, xy, we have one xy. But then we have yx is also the same thing as xy. It doesn't matter what order you multiply it in. So xy plus xy is 2xy. Plus 2xy. And then this last term right here, y times y, that's the same thing as y squared. So we're done. We've used deductive reasoning. We've just used logical steps to start with a statement, to start with an expression really. And we essentially just logically manipulated it. We started with. I guess you could call this a statement-- I guess that's the best thing to call it-- and we logically manipulated to come up with another statement, another fact. We know that this is equal to this using logical properties and distributive property and things like that and properties of exponents.
The content of the passage is a step-by-step explanation of how to use deductive reasoning and the distributive property to prove that \((x + y)^2 = x^2 + 2xy + y^2\).
no_question

the passage does not contain a mathematical question or a word problem. instead, it provides a detailed explanation of the steps to prove an algebraic identity using deductive reasoning and the distributive property.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Course: Algebra (all content)>Unit 18

Lesson 9: Deductive and inductive reasoning

Using deductive reasoning

Sal uses deductive reasoning to prove an algebraic identity. Created by Sal Khan and Monterey Institute for Technology and Education.

Want to join the conversation?

• I got 0=0. Am I doing something wrong here???
(13 votes)
• No. You've just proved the theorem as 0=0 is true.
(37 votes)
• So I was watching these videos to try and gain a fuller understanding of these topics so I could do my homework better. My main struggle was understanding the Law of Syllogism and the Law of Detachment. However, I didn't hear Khan mention these two laws even once during any of these videos. Does anyone happen to know what these two are? If so, would you mind explaining it to me? Or if you heard it during the video at some point, could you tell me where he starts talking about them? If no one else has seen these two laws mentioned, I would like to know why Khan doesn't mention them in here. I can somewhat understand the concepts, but not how to apply it to actual problems. Thank you.
(10 votes)
• You generally will apply these concepts in algebra and geometry. Here's a few examples. The Law of Syllogism states that if we have the statements, "If p, then q" and, "If q, then r", then the statement, "If p, then r" is true. A nice way to conceptualize this is if a = 5, and 5 = b, then a = b. You will use this a lot in traditional geometry proofs. For example, if angle A is congruent to angle B because of the vertical angles theorem and if angle B is congruent to angle C because of the alternate interior angles theorem, then angle A is congruent to angle C because of the Law of Syllogism. You've just proved the corresponding angles theorem. The Law of Detachment states that if we have the statements "If p, then q" and "p" then the statement "q" is true. Another example (picking up from the end of the end of the last example) is, if angle A = 70 degrees, then angle C = 70 degrees because of the definition of congruent and if the measure of angle A = 70 degrees because it is given, then angle C = 70 degrees because of the Law of Detachment. Usually the Law of Detachment is pretty obvious and not directly stated.
(10 votes)
• How do you provide the reasoning for each step?
(0 votes)
• One way to do this is to write the property or definition that justifies each step. Sal talked through the reasoning, but if you have to provide them in written form, put them out to the right side of each step.

Justify (x + y)² = x² + 2xy + y²
(x + y)² = (x + y)(x + y) `definition of exponent a² = a∙a`
(x + y)(x + y) = (x + y)x + (x + y)y `definition of distribution a(b + c) = ab + ac`
(x + y)x + (x + y)y = x(x + y) + y(x + y) `commutative property of multiplication`
x(x + y) + y(x + y) = xx + xy + yx + yy `definition of distribution a(b + c) = ab + ac`
xx + xy + yx + yy = x² + xy + yx + y² `definition of exponent a∙a = a² `
x² + xy + yx + y² = x² + xy + xy + y² `commutative property of multiplication ab = ba`
x² + xy + xy + y² = x² + 2xy + y² `math fact 1 + 1 = 2`
Therefore, we have shown:
(x + y)² = x² + 2xy + y²
using deductive (logical) reasoning, divided into very small logical steps with all the reasoning (justifications) shown.
(9 votes)
• Would it be much easier to use the First Outer Inner Last (FOIL) method once you get to this step:
(x+y) (x+y)
First: x times x = x squared
Outer: x times y = xy
Inner: y times x = yx
Last: y times y = y squared
You would get the exact same thing its just easier to find
(1 vote)
• FOIL is a guideline for beginners who do not yet grasp the distributive property.
All steps in the video are mathematically sound reasoning.
While FOIL gives you the correct answer, it shortcuts the mathematical reasoning - the video is, in a way, proof why FOIL works since you can rearrange, using the commutative and associative properties, the intermediate result Sal gets with what FOIL produces.
You may want to investigate mathematical soundness and validity.

Keep Studying and
Keep Asking Questions!
(5 votes)
• So, to sum it all up, can we say that the mathematical processes including only logical operations come under deductive reasoning and the ones including assumptions come under inductive reasoning?
(2 votes)
• Industive reasoning uses reason, and patterns to come to a conclusion about something, while deductive reasoning uses facts, logic, and definitions to come to a conclusion about something.
(1 vote)
• What type of reasoning would you use to prove a conjecture?
(2 votes)
• Deductive reasoning. A conjecture is something you would form using inductive reasoning. Also, if you prove a conjecture using deductive reasoning, then it becomes a theorem.
(1 vote)
• Okay so Deductive reasoning starts with a fact, while Inductive reasoning is basically using patterns to make a probable solution. am I right?
(2 votes)
• What if the problem is a statement? How do you determine whether its deductive or inductive ?
(2 votes)
• Deduction is drawing a conclusion from something known or assumed. This is the type of reasoning we use in almost every step in a mathematical argument.
Mathematical induction is a particular type of mathematical argument. It is most often used to prove general statements about the positive integers.
So if the problem statement reads, prove that this equals that, it is deductive.
If the problem statement reads, prove that this equals that for all values of n, it is inductive.
(1 vote)
• Can someone answer this question?

If 2x-5=11, then x = 8.
Determine whether the converse is true or false. If true, create biconditional. If false, create a counterexample.

Converse: If x=8, then 2x-5=11.
My teacher said the converse is false because x=8 can be the solution to numerous equations, but if you substitute x=8 to 2x-5, doesn't it equal 11. So, shouldn't the converse be true.
(1 vote)
• If the hypothesis of a conditional is false, can you use the Law of Detachment/Syllogism?
(1 vote)

Video transcript

Use deductive reasoning and the distributive property to justify x plus y squared is equal to x squared plus 2xy plus y squared. Provide the reasoning for each step. Now when they say use deductive reasoning and all this stuff, it might seem like something daunting and new, but this is no different than what we've done in the past. In fact, we've done this very exact problem. So let's just do it step by step, show our logic, and that's essentially deductive reasoning. We're starting with a statement and we're going to deduce that this is going to be equal to something else. So let's start with x plus y squared. We know an exponent means to multiply something by itself that many times. So we know that this is the same thing, or we can deduce that it is the same thing, as x plus y times-- and I'll do this next x plus y in a different color-- times x plus y. That's what x plus y squared is. Now they say to use the distributive property. And this is a bit of review here. And we've seen it many, many, many times before. The distributive property just tells us that if we have a times b plus c-- I want to do as many colors as possible-- that this is equal to a times each of these terms. a times b plus a times c. It's called distributive property cause you're distributing the a in all of the terms in the expression that your multiplying a by. Now we can do the exact same thing here. Instead of an a, you could imagine this is an x plus y. And we can take this entire x plus y and we can distribute it on to both of the terms on this expression that it is multiplying. If this was an a, it'd be ax plus ay. Now that it's an x plus y, we multiply the x plus y times each of those terms. And that's just by the distributive property. So by the distributive property that's going to be equal to-- we'll distribute this on to each of them-- x plus y times x. And actually I don't even have to write the x after it. I could just write it there. It doesn't matter whether you multiply x times x plus y, or x plus y times x. Order doesn't matter. So that's that times that. And then it's going to be plus y times x plus y. And now we can apply the distributive property again. We have x being multiplied by x plus y, then we have a y being multiplied by x plus y. So let's just do that again. So then we get this is equal to x times x plus x times y. I'm going through great pains to keep the colors consistent. Plus y times x. Plus y times this x over here. Plus y times that y over there. I'm doing this a lot slower, and I'm not skipping any steps here. Now what do each of these things equal? x times x. That is the same thing as x squared. So this is equal to x squared. This right here, xy, we have one xy. But then we have yx is also the same thing as xy. It doesn't matter what order you multiply it in. So xy plus xy is 2xy. Plus 2xy. And then this last term right here, y times y, that's the same thing as y squared. So we're done. We've used deductive reasoning. We've just used logical steps to start with a statement, to start with an expression really. And we essentially just logically manipulated it. We started with. I guess you could call this a statement-- I guess that's the best thing to call it-- and we logically manipulated to come up with another statement, another fact. We know that this is equal to this using logical properties and distributive property and things like that and properties of exponents.
user
The content of the passage is a step-by-step explanation of how to use deductive reasoning and the distributive property to prove that \((x + y)^2 = x^2 + 2xy + y^2\).

TYPE: NO_QUESTION

The passage does not contain a mathematical question or a word problem. Instead, it provides a detailed explanation of the steps to prove an algebraic identity using deductive reasoning and the distributive property.