- #1

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Thanks in advance,

Ajay

- Thread starter UncertaintyAjay
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- #1

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Thanks in advance,

Ajay

- #2

WWGD

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- #3

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Or.. you might prove that there are no positive integers ##a,b,## and ##c## such that for ##n \geq 3##

$$a^n + b^n = c^n$$.

- #4

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Haha.

- #5

S.G. Janssens

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What field of mathematics and what level are you interested in?Or even post some theorems to prove?

- #6

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- #7

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You might start by trying to prove some statements about parity. Like an odd number times an odd number is always odd. Even number times anything is always even. Those are pretty straight forward. If you feel you already feel comfortable with those. Then really you should just find a good introductory proofs book, and begin familiarizing yourself with basic proof strategies.

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- #9

S.G. Janssens

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Ok. The following is perhaps more about notation and careful reading than about mathematics proper, but here it is:

Prove or disprove: ##\{x > 0 \,:\, x < r\,\forall\,r > 0\} \neq \emptyset##.

Does that mean that you know what a countable set is?I am also familiar with a few proof strategies like induction

- #10

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I assume this is for x and r both belonging to real numbers?{x>0:x<r∀r>0}≠∅{x>0:x<r∀r>0}≠∅\{x > 0 \,:\, x < r\,\forall\,r > 0\} \neq \emptyset.

Assume the above statement to be true. Then for any element of the set S {r:r>0} there must be another element smaller than it. I.e there is no smallest element of S.( Now i am a bit stuck, I know that there is no smallest element in the set of real numbers, so the statement is true,but I haven't the foggiest idea how to prove it. I'll think over and see if there is another approach to the proof as well)

- #11

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I do. The set of natural numbers,prime numbers, odd numbers, even numbers etc. are countably infinite. Real numbers, irrational numbers etc. are not.Does that mean that you know what a countable set is?

Edit: Sets like "Natural numbers less than 10" are countable. Just not countably infinite. Uncountable sets are ones where you cannot create a one to one function with the set of natural numbers.

- #12

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What about rationals?I do. The set of natural numbers,prime numbers, odd numbers, even numbers etc. are countably infinite. Real numbers, irrational numbers etc. are not.

- #13

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Countable. Right?

- #14

S.G. Janssens

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Yes, that was implicit in the ##>## sign, but you are very correct.I assume this is for x and r both belonging to real numbers?

OkI'll think over

Beautiful, then I already have some other calculus-flavored exercises in mind.I do. The set of natural numbers,prime numbers, odd numbers, even numbers etc. are countably infinite. Real numbers, irrational numbers etc. are not.

- #15

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Bring it on. I haven't been this excited in ages.

- #16

S.G. Janssens

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I was thinking about this one, but it may be a bit too difficult, since a number of steps are required. However, in principle it should be doable.Bring it on. I haven't been this excited in ages.

- Let ##f : [0,1] \to \mathbb{R}## be nondecreasing. Prove that ##f## has a countable number of discontinuities.
- Let ##g: \mathbb{R} \to \mathbb{R}## be nondecreasing. Prove or disprove: ##g## has a countable number of discontinuities.

- #17

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I highly recommend Spivak's Calculus. I found it to be extremely well-written, and fostered a love of pure math in me (an engineering student).

- #18

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Here you can do this problem:

Find the prime number which is one less than a perfect square number?

For more view this link

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