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Write the contrapositive of the following implications: i) If all roses are red, then all violets are blue; ii) If K is closed and bounded, then K is compact_...

Question

Write the contrapositive of the following implications: i) If all roses are red, then all violets are blue; ii) If K is closed and bounded, then K is compact_

Write the contrapositive of the following implications: i) If all roses are red, then all violets are blue; ii) If K is closed and bounded, then K is compact_



Answers

Write (a) the contrapositive and (b) the inverse of each statement.
If those are red and white, then this is blue.

So we have to take the Contra positive of the converse of the inverse of the statement. So start with the anglers. It's not, are not us. Let's go to the converse. If not us, then lot are on that state. The contra positive with this if our witness, we just come back to the same statement.

Hello. Um So part A. Um our um statement is um if I get an A on the final exam I will pass the course. So the converse of that statement is well it's if I pass the course so if I pass the course again this is the converse and first um if I pass the course then um I will get an A on the final exam. So if I pass the course um then I will get N. A. Um I'm the final example so on the final. Okay all right the contra positive, so the contra positive um of our given statement for part A is if I don't pass the course then I will not get an A. So the contra positive is directly um if the statement is true then the contra positive is going to be true as well. The converse on the other hand is not necessarily true. The contra positive here is if I don't pass the course, if I don't pass uh of course mhm. Um Then I will not get an A. On the final. Yeah. Um And then for part B. Um Also our statement for B is if I finish my research paper by friday then I will take off next week. So the converse of that the converse um for part B is if I take off next week. Good. Okay. Mhm. Mhm. Mhm. Um Then I will finish my region so then I will finish um my research papers on my paper. Yeah by friday. It's Mhm. Mhm. All right. And the contra positive for the statement for B is if I do not take off next week then I will not finish my research paper by friday. So again the contra positive is true given that the statement is true. So the contra positive here is if I do not take off next week, I do not take off next week. Mhm. Yeah. Then I will not um finish my hoops, finish my research paper so my paper by friday. Mhm. Mhm. Yeah. Alright. There we have it. Yeah.

Okay. So here um in part a we want to prove that if n is odd then and squared is odd. Well if n is odd then we know that there exists some integer K. Such that N is equal to two K plus one, Right? Because the number is even if it's equal to two K. For some integer K. And if it's odd, well that's one more than an even number. Therefore it's going to be equal to two K plus one. Okay? Um So if you just plug this in for and squared, what do we get? So if n is odd and is equal to K plus one, therefore and squared is going to be equal to two K plus one squared, which is going to be equal to y two K plus one times two K plus one gives us a four K squared plus four K plus one. But do we can actually factor this as the four k squared plus four K. Just factors as two times k squared plus two K. Right. That's too that's um you know, two times two k square. That's gonna be a four K squared plus four K plus one. Um But then we can just go ahead and replace because two K squared plus k, it's just some integer. Just call that some integer. Maybe some integer. I don't know jay. So therefore that n squared is going to be equal to um well just two times some integer called that LBJ. So to J plus one um which shows here that ends squared is just equal to two times interview plus one. Right jay Wright Called J K if you want it doesn't matter what you just equals. En squared is equal to two times the energy plus one. Um This shows that and squared is odd. So therefore if n is odd, we then um conclude that and squared must also be odd as well. Um for part B. Okay, we want to prove this statement. Well, if and squared is odd then N it's going to be odd. Okay, well um we can prove this statement by finding and proving the contra positive, remember that a statement and its contra positive are going to either both be true or both be false. So the contra positive of this statement here um is going to be if um if n is not odd then and squared is not odd. So if n is not odd then and squared is not odd. Yeah or in other words if and if and it's not odd. That means that end is even. So if N is even then and squared is even Mhm Okay. So I have to prove this what we know that if n is even right then and is going to be equal. Um too well some um it's gonna be eager to two K Rachel and is even there exists an integer K. Such that um N is going to be equal to um Well we can write this is actually two K plus two. In other words we can always add to and multiples of two. Um to obtain another even integer. So we plug this in for and squared we then have that and squared is going to be equal to two K plus two squared. Um Which is going two be equal to um well two K plus two times two K plus two is going to be equal to four K squared plus eight K plus four. We can factor the four K squared plus eight K by factoring out um A just a two. So this could be equal to two times and you could go back to two K but only because it's the only factor out A K. We've had four K squared plus eight K is equal to two times two K squared plus four K plus four. Oh but the two K squared plus um two case clippers, four K is just another integer. So therefore therefore we have that and squared is going to be equal to two times. Some introductory call that J lo to J plus four. And we see here that um and squared will also always be even because two times some integer plus plus K plus two times some integer plus four um is always going to be even. So therefore we see that and squared um is also going to be even as well. So therefore we prove the counter positive, which proves the original statement. So I'm gonna conclude here that ends squared and squared is going to be even

Okay. So for um in part a are a given statement is if it rains today then I will stay home from work. So the converse of this is, well it's um if I stay home from work today, so if I stay home from work yeah today yeah um Then it rains. So there is our converse and the contra positive. Well the contra positive um is another way basically the state, the same statement saying in a statement is if it rains today then I will stay home from work. So therefore the contra positive of that is if I don't stay home from work, so if I don't stay home from work um then it does not rain so then it does not um rain today. All right and there is our counter positive. Um For be our given statement is um if the candidate meets all the qualifications then she will be hired. So the converse of this is if the candidate will be hired then she meets all the qualifications. So the converse for B. Is f the candidate yeah um will be hired. Yeah. Then she meets all the qualifications. Yeah. Okay so again there is the converse and then the contra positive is well if the candidate will not be hired then she does not meet all the qualifications. So the converse, right? If the statement is true, the converse um doesn't necessarily is not is not necessarily true. But the contra positive is um is still true to the concert positive. Here is if the candidate if the candidate um will not be hired yeah then she does not. Right. Then we know for sure if she's not hired we know for sure that she um does not meat. Yeah. Article qualifications. Right? This is true given the original statement. So then she does not meet all the qualifications. All right. Mhm.


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