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Which of the following are true? The universe for each is given in parentheses: (a) (Vx)lr +xzx) (Real numbers) (b) (Vx)(x xzx) (Natural numbers) (c) (3x)(2x 3 = 6x...

Question

Which of the following are true? The universe for each is given in parentheses: (a) (Vx)lr +xzx) (Real numbers) (b) (Vx)(x xzx) (Natural numbers) (c) (3x)(2x 3 = 6x 7) (Natural numbers) (d) (3x)(3* 42) (Real numbers) (e) (3x)(3' x) (Real numbers) (T) (3x)(3(2 0= 8(1 x)) (Real numbers) (g) (Vx)Gx2 6x 5 2 0) (Real numbers) (Vx)Gr2 4x 5 > 0) (Real numbers) (3x)Gx? g +41is prime) (Natural numbers) (Vx)62 41is prime) (Natural numbers) (Vx)kr' I7x= 6x + 100 = 0) (Real numbers)

Which of the following are true? The universe for each is given in parentheses: (a) (Vx)lr +xzx) (Real numbers) (b) (Vx)(x xzx) (Natural numbers) (c) (3x)(2x 3 = 6x 7) (Natural numbers) (d) (3x)(3* 42) (Real numbers) (e) (3x)(3' x) (Real numbers) (T) (3x)(3(2 0= 8(1 x)) (Real numbers) (g) (Vx)Gx2 6x 5 2 0) (Real numbers) (Vx)Gr2 4x 5 > 0) (Real numbers) (3x)Gx? g +41is prime) (Natural numbers) (Vx)62 41is prime) (Natural numbers) (Vx)kr' I7x= 6x + 100 = 0) (Real numbers)



Answers

Indicate which of the following are true: (A) All natural numbers are integers. (B) All real numbers are irrational. (C) All rational numbers are real numbers.

In this problem, I'm going to take some statements and use my real number systems diagram to determine if the statements are true we're not. So here's my diagram. I've got natural whole integers rational, irrational and real numbers all in one organizational tool. So this first statement says that all integers are natural numbers. So all integers this box here. Everything inside this box is also inside the natural number box. And this is not true because I've got some integer numbers that are not highlighted double and they would have to be highlighted double to both be true. So this is false. Okay, the second statement, all rational numbers are real numbers. So all rational numbers. Everything inside this big box is also located in this real. That is very true because we have highlighted the rational numbers twice. Okay, our next one is all natural numbers are rational numbers. So all natural numbers. This little box are also located inside the rational number box. And this is also true because our natural numbers are labeled or highlighted twice.

And everyone in this certain your little better human answered my next you are or associative law off my legs in the stairs there for on real number it for my people and see every bracket c medical, er toe a break it busy. So basically, he says, With today why it and so today and I will try to predict food and next wonder Nearby day. Why it? And here today in tow to now they're saying that these two want to be that people in Britain here again for this one and five into trees have been two billion degrees that next you get terrible sits in the fighting. It means associate noise airport amount of Nicoson This water days people their dough this quantity of human ceremonies you Thank you.

You know, there's probably want to classify each of these numbers. Listen them first. If they are a natural number or not, As you may remember, natural numbers are counting numbers. 1234 and then so on. And so, looking through our list here we are looking for counting numbers, and the only county number in that list is 21. And so the only natural number that's 21. Next, we want our whole numbers, and whole numbers are just the natural numbers and zero. And so 21 is definitely a whole number because it's also a natural number. And then in that list of numbers, we also have zero. And so that means that the two whole numbers that we have R zero and 21 next we look for editors and integers are whole numbers and their negatives. Let's say your numbers like this so mhm and so looking here at our list from above, we still have zero and 21 and then we'll also add in negative six because negative sex is a negative natural number. Yes, mhm. Do we have our rational numbers? Mhm. The number is rational. Well, first of all. We've got our fingers, their those will account. So we're gonna put all of those. A number is rational. If it is either a, it is a repeating or terminate industrial so it has a pattern or it stops. Or if it's a fraction and so 5.62 for the 62 repeats that would be rational 0.4, that would be rational. Three and two nights negative. 7/8. Those are all examples of rational numbers. Yeah, on E we want irrational numbers Now. Irrational numbers are just numbers that are not rational, so anything that is in that list that we have not listed already is irrational. Michael Route 23. That's the number that would go on forever and never stopped, as well as 2.74816 because it goes on forever with no pattern, and that's what makes it an irrational number. And then, lastly, on if we want real numbers and real numbers are just all of our numbers here. And so we just list them all into one of six zero. 21 5.62 repeating 0.4 three and two nights. They have 7/8, Route 23 and 2.74816

Hello. So our first statement here is that for every act So this upside down a means for all. So for all Acts X is less than X plus one. Well that is certainly true. Um The reason why I was kind of easy but the reason is that X plus one is always going to be one more than X. Right X plus one is always one more than X. So so yes for any X you add one to it. Well um you get a number which is bigger than that. So access all gonna be less than expose one for any X. That's true. Our next statement. This is gonna be a statement A for statement. Oops after A and then for B. Um we have our statement is there exists a natural number end? So this backwards. E means there exists um a natural number end. So there exists um an end that is a natural number. So an element of this fancy and means the natural number. So there exists an end an element of the natural numbers. In other words, there exists a natural number and um such that all prime numbers. All prime numbers are less than N. Okay, well that statement is definitely false. Um and um well, I mean basically there's a there's a proven good right here. But I mean it's simple proof. I mean, look it up if you don't know it yet, but actually the prime numbers are infinite. So um there exists an end, right? Such that all prime numbers are less than end. Well, okay, we know that the national numbers go on forever. But um the prime numbers, right, go on for ever. Remember you pick there's going to be prime numbers bigger than it. So this statement is false. And then our next statement um is going to be for every X greater than zero. So for all X greater than zero, there exists. Um there exists why? So there exists against backwards evening. There exists um a Y. Such that so such that why is greater than one over X. Um This is actually true. And the reason there is, well, we can certainly pick a number why that is larger than one over X. Um for any value of X that we picked. And our idea our statement is that for every positive number acts. So for all acts greater than zero for any positive acts, there exists a natural number end. So there exists a natural number end. So there exists an element of the natural numbers, there exists a natural number end. Um Such that one over End is going to be less than X. Well, that statement is true, that statement is true and the reason is, well, we I mean the natural numbers are just accounting numbers 1234 and so on. And since end here is in the denominator, we can make the fraction uh one over and here. Um as small as we like, by just changing end to be a larger and larger and larger number. So we can always be able to make one over end. Um less than X, no matter what we pick four X. And our four part E r statement is that for every positive epsilon, epsilon is typically denoted as a very small positive number. So for every positive epsilon, we can regulate this. As for all epsilon, epsilon is a greek letter which looks like a backward street. So for all epsilon um greater than zero. For all positive absalon. So for all epsilon greater than zero, there exists a natural number. End. So there exists again a natural number. End. So there exists an n An element of the natural numbers such that one over to to the end is going to be less than epsilon. Okay, that statement is yes, it is true. And the reason behind this is well, since then um is in the power of to write um in the denominator we can make to to the end as large as we desire, rights similar to that, we can make to to the end as large as we like. Um and therefore one over to to the end can get as small as we desire because it is to to the end gets bigger and bigger and bigger and bigger. One over to to the end get smaller and smaller and smaller and smaller. So one over to to the end can be made as small as we desire by changing end to be a larger and larger number, so therefore we can always be able to make one over to to the end less than epsilon, No matter what positive number we pick for epsilon. Take care.


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