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My teacher solved this using inclusion-exclusion formulas to count the number of surjections from a set of 8 elemets (containing items) to a set of 5 elements (containing boxes). However, I thought of a different solution. But I have a hunch it's wrong. What I thought is to first make sure every...

I am interested in the following theorem: Every field of zero characteristics has a prime subfield isomorphic to ℚ. I am following the usual proof, where we identify every p∈ℚ as a/b , a∈ℤ,beℕ, and define h:ℚ→P as h(a/b)=(a*1)(b*1)-1 (where a*1=1+1+1... a times) I have worked out the...

Let (A,+) be an Abelian group. Consider the ring E=End(A,A) of endomorphisms on the set A, with binary operations +, and *, where (f+g)(x)=f(x) + g(x), and (f*g)=f∘g. I have tried to find zero divisors in this ring, but I just couldn't come up with an example.

I gave little information, and I am sorry. To skip the settings, here's straight to the problem. Say we want to prove that limit of the function f(x)=sinx/x as x approaches 0 is 1. We can play around and get that cosx<sinx/x<1 for 0<x<π/4. Since the limit of cosx as x approaches 0 is 1, and...

When we define a limit of a function at point c, we talk about an open interval. The question is, can it occur that function has a limit on a certain interval, but it's extension does not? To me it seems obvious that an extension will have the same limit at c, since there is already infinitely...

So, I recently came across this example: let us "define" a function as ƒ(x)=-x3-2x -3. If given a matrix A, compute ƒ(A). The soution proceedes in finding -A3-2A-3I where I is the multiplicative identity matrix. Now , I understand that you can't add a scalar and a matrix, so the way I see it is...

So is my 2nd post incorrect? I know that the given projection is not injective, but it is surjective. Can we then restrict it to an injective one, and get a new function from a subset of the domain to the codomain? From here it follows that the second set is either finite or countable. Anyways...

The book by K.Kuratowski, A.Mostowski, Set Theory, is mentioned in the bibliography.

This book is in Croatian, and you can say it is not really a book, more like a compilation of notes made by one of our professors. Also, sorry for posting twice. I didn't know about the convention.

So, I thought about this, and this is what I have concluded. Since there is a surjection from ℤxℤ* to ℚ, then there is injection from S⊂ℤxℤ* to ℚ, which means that there is a bijection from S to ℚ. Since ℤ⊂ℚ, ℚ is infinite, but then S is infinite too. Since S is an infinite subset of a countable...

I know there are many proofs of this I can google, but I am interested in a particular one my book proposed. Also, by countable, I mean that there is a bijection from A to ℕ (*), since this is the definition my book decided to stick to. The reasoning is as follows: ℤ is countable, and so iz ℤxℤ...

This may be a silly way to approach it, but I thought of this. (∀n∈M) s(n)∈M is by definition equivalent to (∀n)(n∈M →s(n)∈M), which is obviously not equivalent to (∀n∈ℕ)(n∈M →s(n)∈M). Another way to think about it is that these can become equivalent if we consider a few things, which is not...

So , what I was wondering about was a slight difference in notation, for which I am not certain if correct (mine, in particular.). The induction axiom says: If M is a subset of ℕ, and if holds that: a)1∈M b)(∨n∈ℕ)(n∈M→s(n)∈M) then M=ℕ. Now my question is: why do we write (∨n∈ℕ)(n∈M→s(n)∈M)...

There is nothing wrong, sorry. I overlooked it.

I used wrong to derive wrong, intending to show that the former had to be wrong. That is, I tried to find a contradiction.

I meant that √f(x) > g(x) is equivalent to f(x)≥0 ∧ g(x)≥0 ∧ f(x)>(g(x))^2. Sorry.

Also, if √f(x)> g(x), and if f(x)≥0 and g(x)>0 and f(x) > (g(x))^2 then f(x) can = 0 , but 0 is not greater than any g(x) > 0, so f(x) should be strictly greater than zero.

Does that mean that √f(x) < g(x) is not equivalent to f(x)≥0 ∧ g(x)≥0 ∧ f(x)>(g(x))^2, but to f(x)≥0 ∧ g(x)>0 ∧ f(x)>(g(x))^2, or both, since f(x)>(g(x))^2 implies that g(x) cannot be equal to zero, so it is the same thing? I don't understand then why would anybody write g>0 for √f(x)<g(x) and...

So, I know that the inequality √f(x)<g(x) is equivalent to f(x)≥0 ∧ g(x)> 0 ∧ f(x)<(g(x))^2. However, why does g(x) have to be greater and not greater or equal to zero? Is it because for some x, f(x) = g(x)=0, and then > wouldn't hold? Doesn't f(x)<(g(x))^2 make sure that f(x) will not be...

Thank you, I understand now.

What I am asking, and sorry for being ambiguous, is, if we have two functions f and g, and if the composition gof is defined (we don't need restrictions), in this case, is the domain of gof the domain of f, and the codomain of gof, the codomain of g. Thanks.

What happens with the codomain? Does it remain the codomain of f?

So, I'm a bit confused. The thing is, basically, all elementary functions are of the form ƒ:ℝ→ℝ. So the domain is ℝ and so is the codomain. However, if we have a function ƒ:ℝ→ℝ, given with f(x) = √x, it's domain is now x≥0. So, is the domain of this function ℝ or [0,+∞>? Also, let's say we have...

Thank you.

Also, is saying there exists A with property B, the same as, for some A holds the property B?

So, then, should the statement (∀x∈A)(∃y∈A)((x,y)∈R) , where A is the empty set and R a relation on the empty set (hence, empty relation), also be false? Or do we ignore everything that comes after ∀x∈A and consider the part (∃y∈A)((x,y)∈R) as some P(x,y)?

So, here's my question. I read somewhere that all universal truths on empty domains are vacuously true, whereas all existential are false. However, if all statements of the form (∀x∈A)(P(x)) , where A is an empty set, are vacuously true, then the statement (∃x∈A)(P(x)) should also be true...

Wow, an excellent answer! Thank you...

You are right, I said smaller, but meant greater. Sorry.

Thank you. :)

Not sure if a compliment or a reference that I do not really understand the subject.

What I meant by smaller is n<x. English is not my mother tongue and I am probably not using words the way they are supposed to be used.

Right, so "exists" stays with the n and "all" stays with x, they don't swap places? Thank you anyway.

Hmmm, but here we have another variable n that that also has some property... How to include this? Sorry if I don't understand.

But isn't that exactly the negation of the statement, not the converse? I am interested in "if Q, then P" if the given statement is "If P, then Q", even though this statement isn't in the if - then form. Sorry if I didn't understand you.

The sentence is : "For all real numbers there exists a natural number that is smaller". That is (∀x∈R)(∃n∈N)n>x. This is what I thought of: we can write this sentence as:"If x is a real number, then there exists a natural number n that satisfies n>x." So how would I make a converse statement...

That is what I was thinking. So basically, if we say, for example, show that something doesn't hold universally, our task is to disprove an universal statement, that is to prove the negation of the statement by giving an example. However, this still has some connection to the original statement...

Does a proof by counterexample belong to direct or indirect type of proof?

Wow, I would've never thought of something like that. Although this way of proving is a little too complex for me. Thank you for your reply!

Thank you RUber!

The problem: prove that if 1/a + 1/b + 1/c=1 and a,b,c are positive numbers then (a-1)(b-1)(c-1)>=8 I've tried it myself but couldn't do it. I have tried going backwards i.e. : (a-1)(b-1)(c-1)>=8 (fast forward) abc -(ab+bc+ac) +a+b+c -1 >=8 Now from 1/a + 1/b + 1/c = 1 we have that...

First, thanks for your replies! I actually live in a second world country so the curriculum is very different from that of the west.In the first two years of HS we studied maths the way gymnasiums do it, i.e. we learned algebraic fractions,factoring,potencies, some analysis , geometry, then...

I am in the 3rd grade of high school and we have a very weird math program.Since the school is specialised for economics we don't study trigonometry in the 3rd grade ,instead we learn about interest rates and how to calculate credits etc... Regardless, we have additional lections (which I am...