WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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This next lesson is on deductive reasoning.
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**Deductive reasoning** is the process of reasoning logically--that is the keyword right here, "logically."
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You are going to use logic from given statements to form a conclusion.
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If given statements are true, then deductive reasoning produces a true conclusion.
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As long as we have statements that we can show as true, then based on those statements, we can come to a true conclusion.
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And this is the process of deductive reasoning.
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Many professions use deductive reasoning: doctors, when diagnosing a patient's illness...
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A few lessons ago, we learned about inductive reasoning; that is the opposite of deductive reasoning.
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Inductive reasoning uses, remember, examples and past experiences.
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But for deductive reasoning, each situation is unique, and you are going to look at basically facts and truths--
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anything that is true--to come up with that conclusion.
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Doctors, when you diagnose a patient's illness, have to look at all the facts and what is there to be able to diagnose the illness correctly.
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You don't want doctors to diagnose based on inductive reasoning, because then, as long as you have the same symptom, then you have the same illness.
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For example, if you have a bruise, and you go in to see the doctor, inductive reasoning would suggest that,
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well, since the last two patients that came in with bruises had some sort of illness,
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you will have the same illness also, just because you have a bruise.
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But deductive reasoning...again, you have to look at each unique situation, and looking at that individual,
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and all of the given statements, all of what is true, the facts there--using that, the doctors will diagnose the patient's illness.
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Carpenters, when deciding what materials are needed at a worksite: each time a carpenter has a different site, they need a different material.
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So, deciding what materials to use at that specific worksite is considered deductive reasoning.
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Again, inductive reasoning is using examples, past experiences, and patterns to make conjectures.
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You make conjectures; you make guesses, using "Well, it happened this way the last five times,
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so the sixth time, I can make a conjecture that it is going to happen again."
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So, a conjecture is an educated guess.
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Now, with deductive reasoning, you use logic, and you use rules, to come to a conclusion.
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With inductive reasoning, you are just kind of guessing, just by patterns, what is going to come up next.
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But with deductive reasoning, you are actually looking at the situation, and you are going to use logic;
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and you are going to use rules and facts to make a conclusion, to base it on something.
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The first law of logic is the Law of Detachment.
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Now, if a conditional is true, and the hypothesis is true, then the conclusion is true.
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If you look at this, it will be easier to understand.
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This is the conditional statement: if p → q is true, and p is true, then q will be true.
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As long as the conditional is true and the hypothesis is true, then the conclusion will be true.
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Here is an example: If a student gets an A on the final exam, then the student will pass the course.
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That is the conditional p to q: If a student...here is p; all of this is p, "a student gets an A on a final exam."
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Then, the student will pass the course; here is q; so p to q is true.
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Now, David gets an A on the geometry final; here, this is this p, so that is true,
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because the conditional statement says that if a student gets an A on the final exam, then the student will pass the course.
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Well, David got an A on the final exam; then what can you conclude--what kind of conclusion can you make?
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It is that David, then, will pass the course.
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So, this conditional was true; "If a student gets an A on a final exam, then the student will pass the course"--that is the given conditional.
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Then, David gets an A on the final exam; that is part of this.
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So, if he gets an A on the final exam, then you can say that he is going to pass the course, because that is what the conditional says, and the conditional is true.
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The next example: If two numbers are odd, then their sum is even.
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Two numbers are odd--here is p; their sum is even--here is q.
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And then, 3 and 5 are odd numbers; this is based on p--this is all based on p.
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p → q is true, and this right here, "3 and 5 are odd numbers"...then my conclusion is that the sum of 3 and 5 is even.
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The sum is going to be even, then, because this is the conditional.
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If two numbers are odd, then their sum is even; and 3 and 5 are odd numbers; then, the sum of 3 and 5 is even.
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You are using the conditional and a hypothesis; then you are going to come to a conclusion.
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And this is the Law of Detachment: if p → q is true, and p is true, then q is true.
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The next one is the Law of Syllogism; this one is very similar to the transitive property of equality.
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If you remember, from Algebra I, you learned the transitive property.
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The transitive property says that, if A equals B, and B equals C, then A equals C.
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If A equals B, and B equals C, then, since these two are equal, A equals C.
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This is very similar to that: the Law of Syllogism says that if the conditional p → q is true,
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and q → r, that conditional, is true, then p → r is also true.
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So then, here you have two different conditional statements.
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You have p → q, and then you have q → r; now remember, this q and this q have to be the same.
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p → q is true; q → r is true; this is a different conclusion; then, this hypothesis, p, to this conclusion, r, is going to also be true, just like this one.
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A to B and B to C...then A is equal to C.
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Let's just do a couple of examples: Using the two given statements, make a conclusion, if possible.
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If M is the midpoint of segment AB, then AM is equal to MB.
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If I have segment AB, and M is the midpoint (this is M), then AM is equal to MB.
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If the measures of two segments are equal, then they are congruent.
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Here, this segment and this segment are equal; right here, that is what it says.
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Here is AM, and here is MB, and they are equal to each other; then, they are congruent.
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So, all of this right here--this is all p; this first one would be p → q; and then, this is q.
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This next one, "The measures of two segments are equal," is saying the same thing as this right here: AM = MB/two segments are equal.
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So then, this is using q; then they are congruent--now, this is a new conclusion, so this is r; so this is q → r.
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So, my conclusion...see, right here, the Law of Syllogism says p → q; there is p → q;
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then q → r--this q → r; then p → r is also true.
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So, I can come up with a true conditional statement by using this.
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Then, I can say that my p is here; so if M is the midpoint of segment AB, then the segments are congruent.
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And I can also say "then AM is congruent to MB," because this one uses AB, so I can just say AM is congruent to MB.
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I can write it like this, or I can write it like this.
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Here, I used p; all of this is p; and then, the segments are congruent, so that is r; so this was p to r.
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So, since this is true, and this is true, then this is what I can conclude: p → r is also true, by the Law of Syllogism.
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Let's do the next one: If two angles are vertical, then they do not form a linear pair.
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Here is p; then they do not form a linear pair--this is q; this one is p → q.
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Then, if two angles are vertical--look at this one--this is the same as right here; so this one is p.
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Then, they are congruent; this is r; so this is p → r.
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Well, here I have p → q; and the Law of Syllogism says that p → q and q → r have to be true.
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I can't have p → q and then p → r; I can't come up with a true conclusion, because here it is not q → r; it is p → r.
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With this, I can't form a conclusion; so this one is no conclusion.
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Let's do a few examples: we are going to use the laws of logic, the ones that we just learned, the Law of Detachment and the Law of Syllogism.
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And we are going to determine if statement 3, the third statement, follows logically from true statements 1 and 2.
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Based on the first one and the second one, we are going to see if the third one is going to be a true conclusion.
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Number 1: Right angles are congruent--that is the first statement.
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Now, this is not written as a conditional; so if you want, you can rewrite it as a conditional.
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Or you can just remember that this part right here is going to be the hypothesis, and this part right here is going to be the conclusion.
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I will just write out the conditional of "right angles are congruent": "If the angles are right angles, then they are congruent."
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And that is the congruent sign; if angles are right angles, then they are congruent.
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Now, it is easier to see that this is my hypothesis; that is p; and "they are congruent"--this is q.
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This first one was p → q; now, the second statement is "Angle A and angle B are right angles."
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Here we have right angles; now, do we see that?--that sounds familiar to me.
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It is right here; the angles are right angles; so angles A and B are right angles.
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This is p; or we can write it here--p.
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Then, angle A is congruent to angle B: is that the correct conclusion?
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Well, here, if the angles are right angles, and it says that angle A and angle B are right angles, then what?
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They are congruent; so then, this says that they are congruent; so this is q.
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This is true; this is a valid conclusion, based on the Law of Detachment, because the Law of Detachment says that,
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if p → q is true, and p is true, then q is true.
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So, it is valid; this is the Law of Detachment.
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The next one: Vertical angles are congruent.
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Vertical angles: this one is p; them being congruent: that is q, so this one is p → q.
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Angle 1 is congruent to angle 2--now, is that from p or q?
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That is from q, because it says that angles are congruent here; so this one is q.
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And angle 1 and angle 2 are vertical angles--this is p...this is actually...I wrote p instead of q right here.
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This one is q, and the conclusion was that angle 1 and angle 2 are vertical angles, which is p.
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Now, we don't have a law of logic that says that if p → q is true, and q is true, then p is true.
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That is not any law; it looks like the Law of Detachment, but the Law of Detachment is that if p → q is true, and p is true, then q will be true.
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OK, so in this case, this is an invalid conclusion.
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See, p → q and p--then q will be true; it can't be the other way around.
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This is invalid; this is actually the converse, and that is not true.
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Again, using the Law of Detachment and the Law of Syllogism, determine if statement 3 follows logically from true statements 1 and 2.
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And state which law is used.
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The first one: inline skaters live dangerously: here, "inline skaters" would be p; they "live dangerously"--that is q.
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"If you live dangerously"...that is the same thing as q; so this is q..."then you like to dance"; this is a new statement, so this is r.
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"Inline skaters"--this is p--"like to dance"--this is r.
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So, here "inline skaters live dangerously" is p → q; this is q → r; then the third statement,
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"inline skaters like to dance"--this is p → r; and this is valid by the Law of Syllogism.
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This one says, "Inline skaters live dangerously"; that is p → q.
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If you live dangerously--that is the same statement as this one right here--then you like to dance; that is q → r.
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So, this one right here and this one right here are the same.
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"Inline skaters"--that same statement right there is p--"like to dance"--that is r.
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And the Law of Syllogism, remember, says if p → q is true, and q → r is true, then p → r is true.
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So, it is like the transitive property--the Law of Syllogism.
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The next one: "If you drive safely, the life you save may be your own."
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Here, this is p; "the life you save may be your own"--here is q.
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"Shani drives safely"--that is from p; "the life she saves may be her own"--this one is q.
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This is the same as this one; so the first statement is p → q, and the next one is p; the conclusion,
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"the life she saves may be her own," is q; so based on 1 and 2, based on these two, we are able to get this.
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Yes, this is valid; and this is one is by the Law of Detachment.
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The Law of Detachment says that if p → q is true, and p is true, then q is true.
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We are going to do a few more examples: the first one: Draw a conclusion, if possible; state which law is used.
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If you eat to live, then you live to eat: "If you eat to live"--this one is p--"then you live to eat"--that is q; that is p → q.
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"Christina eats to live": that is from statement p, so draw a conclusion.
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Our conclusion is, then, "Christina lives to eat," because if p → q is true, and p is true, then I can conclude that q is true.
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And this one was by the Law of Detachment.
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The next one: "If a plane exists, then it contains at least three points not on the same line."
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"If a plane exists"--there is p--"then it contains at least three points not on the same line"--there is q.
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And to draw this out: this is just saying that if I have a plane, then contains at least three points in the plane that are not on the same line.
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Plane *N* (let me draw plane *N*--here is plane *N*) contains points A, B, and C, which are not on the same line.
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If a plane exists, then it contains at least three points not on the same line.
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Plane *N* contains points A, B, and C, which are not on the same line.
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Well, all of this right here is from statement Q; we have...so I have p → q, and then I have a q.
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So, I cannot come to a conclusion; I cannot draw a conclusion, because there is no law that says that if p → q is true, and q is true, then p is true.
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So, I can't say, "Then plane *N* exists"--that is not a conclusion that I can come to.
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In this case, my answer will be no conclusion--it cannot be done.
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Draw a conclusion, if possible; state which law is used.
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If you spend money on it, then it is a business; if you spend money on it, then it is fun.
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Let's label these: this right here is p; "then it is a business" is q; so this is p → q.
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"If you spend money on it"--well, that is p; "then it is fun"--this is r; it is not the same as q, so it is r; this is p → r.
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Now, can I draw a conclusion based on these statements?--no, because there is no law that says that,
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if p → q is true, and p → r is true, then q → is true.
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It has to be p → q; so the Law of Syllogism says p → q and q → r; then p → is true--this is the Law of Syllogism.
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So, in this case, since it is p → q and p → r, this has no conclusion.
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The next one: if a number is a whole number, then it is an integer.
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Remember that whole numbers are numbers like 0, 1, 2, 3, and so on.
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And integers are whole numbers and their negatives, so it is going to be -2, -1, 0, 1, 2, and so on; those are integers.
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"If a number is a whole number"--there is my p--"then it is an integer"--there is a q.
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"If a number is an integer"--isn't this q?--"then it is a rational number"--this is r.
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And rational numbers are numbers that are integers (it could be -2); I can have fractions; I can have terminating decimals--all of that.
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Now, remember: these given statements are true statements, and you are trying to see if you can use those true statements to draw a conclusion.
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Here is p → q; this one is q → r; remember: if we have p → q and q →, then we can say that p → r is true.
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That is going to be our conclusion: p → r; so I can say, "If a number is a whole number, then"--here is p--
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I am going to draw my conclusion, p → r--"then it is a rational number."
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All of this is p, and all of this is r; and that would be valid because of the Law of Syllogism.
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The next example: Determine if statement 3 follows logically from statements 1 and 2; if it does, state which law is used.
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Based on 1 and 2, we are going to see if number 3 is valid.
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If you plan to attend the university of Notre Dame, then you need to be in the top 10% of your class.
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Here is my p; "you need to be in the top 10% of your class"--there is my q; so this one is p → q.
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Jonathan plans to attend Notre Dame; so this one is p.
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Jonathan needs to be in the top 10% of his class; this is q--yes, that is q--"then he needs to be in the top 10% of the class."
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Based on these two, numbers 1 and 2, these are true statements; statement 1 is true, and statement 2 is true.
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Then, is my conclusion, my statement 3, true?--yes, this statement is valid, because of the Law of Detachment.
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The next example: Determine if statement 3 follows logically from statements 1 and 2.
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If it does, state which law was used.
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We are going to see, again, if the third statement is valid or invalid, based on these two true statements.
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So, if an angle has a measure less than 90, then it is acute.
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"An angle has a measure less than 90"--that is my p; then "it is acute"--this is q; so my conditional is p → q.
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"If an angle is acute"--well, isn't that what this is right here?--so here is q.
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"Then its supplement is obtuse"--the supplement is an angle measure that makes it 180.
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So, if we have two supplementary angles, then it is two angles that add up to 180.
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A supplement of an angle would be the number, the angle measure, that you would have to add so that it would add up to 180.
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Then, it is obtuse; this is r; this is the new statement, so here we have q → r.
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"If an angle has a measure less than 90"--here is my p; all of this is p--"then its supplement is obtuse"--this is r.
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So, this, my third statement, was p → r; well, does that follow any rule, any law?
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p → q is true; q → r is true; then p → r is true; so this is valid, and it is from the Law of Syllogism.
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And the next example: If a figure is a rectangle, then its opposite sides are congruent.
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If I have a rectangle, its opposite sides are congruent; so this is congruent to here, and this is congruent to here.
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AB is congruent to DC; so if I have ABCD, AB is congruent to DC, and AD is congruent to BC.
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The figure is a rectangle--there is my p; then its opposite sides are congruent, so there is q.
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AB is congruent to DC, and AD is congruent to BC.
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Here is q, because it says "if the opposite sides are congruent"; ABCD is a rectangle; this is p.
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I am going to use a different color for that one; this is p.
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So, statements 1 and 2 are p → q, and statement 2 is q, and my third statement is p.
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Can you use these two to make this conclusion, that p is true?--no, so this is invalid.
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This statement right here is invalid; the Law of Detachment says, if p → q is true, and p has to be true, then q is true, not the other way around.
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This one is invalid; make sure that this second statement has to be p, and then your conclusion is going to be q.
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OK, well, that is it for this lesson; thank you for watching Educator.com.