All Questions
9
questions
2
votes
1
answer
33
views
Equivalent characterizations of finite sets
How can we show that the following notions of finiteness for a nonempty set $X$ are equivalent?
There exists $n \in \mathbb{N}$ such that there is an injection $X \hookrightarrow \{1, \ldots, n\}$
...
- 1,649
4
votes
0
answers
115
views
Show that $f(a,b,c)=(a+b+c)^3+(a+b)^2+a$ is injective.
For a function $f : \mathbb{N} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ defined by
$f(a,b,c)=(a+b+c)^3+(a+b)^2+a$
I want to show that $f$ is injective.
How can I show this?
I ...
- 41
3
votes
1
answer
490
views
Is this exercise from Tao's Analysis 1 erroneous?
On page 68 of the fourth edition of Tao's Analysis 1, is Exercise $3.5.12$, the first part of which I believe is erroneous. The exercise is stated as follows:
(Note: $n++$ refers to the successor of $...
- 469
1
vote
1
answer
25
views
Function that generates 2 indexes for nested subsets
I'm doing a problem and looking for some way to create a specific bijection between $\mathbb{N}$ and my set $W$. $W$ is a set that contains infinite subsets $Y_n$. Each of these subsets contains every ...
- 121
3
votes
2
answers
140
views
How do I prove injective property of $(x + y)^2 + y: \mathbb{N}×\mathbb{N} \to \mathbb{N}$
Given this function: $(x + y)^2 + y$, how do I go about proving it's injective property of mapping $\mathbb{N}×\mathbb{N} \to \mathbb{N}$ ? Surjection is not required. My current attempts include ...
- 59
0
votes
1
answer
80
views
Is this proof using Cantor's Diagonal Argument correct?
We use $\sim$ to indicate to sets being bijective to eachother, i.e. having the same cardinality in this context. There exists $\psi:\mathbb{N}^2\mapsto\mathbb{Q}$ ,via $\left(a,b\right)\mapsto\frac{a}...
- 650
0
votes
1
answer
142
views
Show that the cardinality of ($X$ ∪ {$x$}) is equal to the cardianlity of ($X$)+$1$
Let $X$ be a finite set, and let $x$ be an object which is not an element of $X$. Then $X$ ∪ {$x$} is finite and #($X$ ∪ {$x$}) = #($X$)+$1$. Note that #-here means cardinality.
Suppose the cardinality ...
- 1,689
1
vote
1
answer
648
views
Show that the finite subsets of the natural numbers are bounded.
Let $n$ be a natural number, and let $f : ${$i ∈ N :1≤ i ≤ n$}→$N$ be a function. Show that there exists a natural number $M$ such that $f(i) ≤ M$ for all $1 ≤ i ≤ n$.Thus finite subsets of the natural ...
- 1,689
1
vote
1
answer
313
views
How to define addition on the naturals as a function rather than with the successor "function".
It is my understanding that when an algebraic structure (in this case a commutative monoid) is equipped with an operation (in this case $+$) that binary operation is a function of two variables.
I ...
- 701