help with writing maths

I need help with a few questions. I've copied the questions and my attempts at them.

1) Turn words into symbols, using standard or Zermelo definitions (or any other symbolic representation).

•

The set of rational points in the open unit cube

I've done this:

\{(x,y,z)\in\mathbb{Q}: 0<x,y,z<1\}

but it looks wrong to me.

2) For each item, provide
(i) a coarse description, which only identiﬁes the class to which
the item belongs (set, function, polynomial, etc.);
(ii) a ﬁner description, which deﬁnes the object in question, or
characterises its structure

•

$(1, 1 + x, 1 + x + x^2, 1 + x + x^2 + x^3, . . .)$

sequence, an infinite sequence with increasing terms (is this the right way of wording this?!)

•

13 = (3 + 2√−1)(3 − 2√−1) Is this just a quadratic expression?

•

$\sum\limits_{k=1}^\infty \frac{1}{k^2 + x^k}$

an infinite sum of fractions with increasing degree...?

Sorry for all the questions. A bit of help would be appreciated!

Reply 1

ghostwalker

Original post by #maths

•

The set of rational points in the open unit cube

I've done this:

\{(x,y,z)\in\mathbb{Q}: 0<x,y,z<1\}

but it looks wrong to me.

A rational point is a point whose coordinates are rational. The ordered triplet is not a rational number.

Have another go.

(edited 10 years ago)

Reply 2

#maths

Original post by ghostwalker

A rational point is a point whose coordinates are rational. The ordered triplet is not a rational number.

Have another go.

I know a point with rational coordinates would be

z=(x,y)\in Q^2

and the unit cube is

[0,1]^3

but I don't know how to put them together,

the set of rational points in the open unit cube is

\{z \in Q: 0<z<1\}

this is obviously wrong :frown:

(edited 10 years ago)

Reply 3

ghostwalker

Original post by #maths

I know a point with rational coordinates would be

z=(x,y)\in Q^2

As you noticed the ordered pair (x,y) is not a rational number, it cannot be an element of Q.

BUT it can be an element fo Q^2 - as you posted.

So, just extend that to three coordinates. You're only missing the exponent in your original post.

(edited 10 years ago)

Reply 4

BlueSam3

Original post by #maths

I know a point with rational coordinates would be

z=(x,y)\in Q^2

and the unit cube is

[0,1]^3

but I don't know how to put them together,

the set of rational points in the open unit cube is

\{z \in Q: 0<z<1\}

this is obviously wrong :frown:

That's the one dimensional version (the open unit interval

(0,1)

). You have already posted the two dimensional version (the open unit square (0,1) \times (0,1)) You want the three dimensional analogue. You were very close at the start.

Reply 5

maths2015

We're struggling with the same questions this year. Can you help us please?

Reply 6

ghostwalker

Original post by maths2015

We're struggling with the same questions this year. Can you help us please?

Post details of what you've done/tried and where you're stuck.

Is this correct?

Original post by maths2015

Is this correct?

Nopes, why would

(x, y, z) \in \mathbb{Q}^2

be true? Wrong exponent.

Reply 9

maths2015

Would it be Q^3 instead?

Reply 10

Zacken

Original post by maths2015

Would it be Q^3 instead?

Yups!

Reply 11

maths2015

Thank you so much! :smile:

Reply 12

maths2015

Another question: {x∈R\Q : x2 ∈N} in words.

* Set x is a set of irrational numbers such that x^2 are natural numbers.
Is this correct?

Reply 13

Zacken

Original post by maths2015

Another question: {x∈R\Q : x2 ∈N} in words.

* Set x is a set of irrational numbers such that x^2 are natural numbers.
Is this correct?

I don't see anything wrong with it.

Reply 14

maths2015

Question:
For each item, provide two levels of description: (i) a coarse de- scription, which only identifies the class to which the item belongs (set, function, polynomial, etc.); (ii) a finer description, which defines the object in question, or characterises its structure