All Questions
16
questions
17
votes
8
answers
7k
views
Can we have a one-one function from [0,1] to the set of irrational numbers?
Since both of them are uncountable sets, we should be able to construct such a map. Am I correct?
If so, then what is the map?
- 6,016
6
votes
3
answers
159
views
Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ that satisfies these conditions?
Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ from the reals into the power set of the naturals such that
for any $x\in\mathbb{R}$ the set $f(x)$ is infinite, and
for any distinct $x,...
- 63
3
votes
1
answer
4k
views
Bijection from the irrationals to the reals
Since the irrationals and the reals have the same cardinality, there must be a bijection between them. Somewhere on this forum I found something like this:
Map all of the numbers of the form $q + ...
- 3,719
2
votes
1
answer
61
views
Define powerset P(f) (P(real numbers)
I'd like some help for clarification, as I have no professional help to ask (and also wouldn't want to pay for it yet).
This is part of a German book on mathematical analysis, I don't want the ...
- 85
1
vote
1
answer
115
views
Can construct a bijection between R - Q and ( (R - Q) ∩ [0,1] )?
I've tried to show that:
$$[0,1]\sim([0,1] ∩R-Q)$$
I know from this answer :
$$[0,1]\sim R-Q$$
But how to construct a bijection between R-Q and $([0,1]∩R-Q)$ ?
I think the function would be like $f:R-...
- 382
1
vote
2
answers
231
views
Set of constant functions are uncountable.
Let $F=$ $\{$ $f: [0,1] \rightarrow \mathbb{R}$ $:$ $f$ is constant$ \} $. I must show that $F$ is uncountable.
Note, that for any $f \in F$, and any $c\in \mathbb{R}$, I will denote the constant ...
user643073
1
vote
0
answers
70
views
Is this a surjection from $(0,1) \rightarrow \mathbb{R}$?
I am trying to think of creative bijections from $(0,1) \rightarrow \mathbb{R}$ that can serve as a proof of the fact that the cardinality of the interval $(0,1)$ is equal to the cardinality of $\...
- 1,186
1
vote
0
answers
44
views
Question concerning defining a particular class of functions
I have a multiset of real numbers $X \subseteq \mathbb{R} $ and I want to create a class of injective function to map the elements of $X$ to the unit interval(so basically a normalization).
However ...
- 75
0
votes
1
answer
62
views
Proving that a specific function $2^{\mathbb{N}}\to\mathbb{R}$ is injective.
Consider $\mathbb{R}$ as a set of Dedekind cuts. $A \subseteq \mathbb{Q}$ is a Dedekind cut if
$A \notin \{\varnothing,\mathbb{Q}\}$,
$(\forall q \in A)(\forall p \in \mathbb{Q})(p < q \...
- 4,528
0
votes
1
answer
827
views
Does $f:\mathbb{R} \rightarrow \mathbb{R}$ mean that $f(x)$ is defined for all real inputs?
$f:\mathbb{R} \rightarrow \mathbb{R}$ basically means that the domain of $f$ is the set of real numbers and the range of $f$ is the set of real numbers. However, does it mean that $f(x)$ is defined ...
- 3,186
0
votes
1
answer
124
views
Clarification on intuition behind one to one correspondence?
My book - Discrete Mathematics and its Applications
This is my book's definition on if an infinite set is countable
And the example it gave
The "infinite set is countable if and only if it is ...
- 1,884
0
votes
0
answers
57
views
Finding functions that intersect at the minimum number of points
Let $f$ and $g$ be (non-constant) functions from $\mathbb{R}^d$ to $\mathbb{R}$.
For a point $x \in \mathbb{R}^d$, let us define the set $S_{f,x} = \{y \in \mathbb{R}^d : f(y) = f(x) \land y \neq x \}$...
- 33
0
votes
0
answers
103
views
Domain of function $y$
In my physics book I saw the following math snippet:
Let
$$y(t)=\sin(t)\int_{-\epsilon}^{\epsilon}x(\tau)\cos(t-\tau)d\tau$$
be the output signal for input signal $x(t)$.
So, as a ...
user503399
0
votes
0
answers
42
views
Proving that a certain set of sequences is uncountable
Let $B:=\{(b_1, b_2, b_3, \ldots) : b_i =\pm i!$ for every $i \in \mathbb{N}\}$.
I WTS that $B$ is uncountable. I know there are several ways to do this. At this point I think that constructing a ...
- 4,174
-1
votes
1
answer
44
views
Does there exist a real-valued function such that for any proper subset of real numbers the function maps outsidr of the set?
Does there exist a bijective function $f$ from the real numbers to the real numbers such that for any non-empty proper subset of real numbers $R$ there exist $x$ in $R$ such that $f(x)$ is not an ...
- 1,104