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Omni-MATH / [b]a)[/b] Is it possible to find a function f:mathbb N^2tomathbb N such that for every function g:mathbb Ntomathbb N and positive integer M there exists ninmathbb N such that set …

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problem

[b]a)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ and positive integer MM there exists $n\in\mathbb N$ such that set $\left\{k\in \mathbb N : f(n,k)=g(k)\right\}$ has at least MM elements? [b]b)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ there exists $n\in \mathbb N$ such that set $\left\{k\in\mathbb N : f(n,k)=g(k)\right\}$ has an infinite number of elements?
Plain-text mathematical notation (without MathML)
[b]a)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ and positive integer M there exists $n\in\mathbb N$ such that set $\left\{k\in \mathbb N : f(n,k)=g(k)\right\}$ has at least M elements?
[b]b)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ there exists $n\in \mathbb N$ such that set  $\left\{k\in\mathbb N : f(n,k)=g(k)\right\}$ has an infinite number of elements?
Original LaTeX notation
[b]a)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ and positive integer $M$ there exists $n\in\mathbb N$ such that set $\left\{k\in \mathbb N : f(n,k)=g(k)\right\}$ has at least $M$ elements?
[b]b)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ there exists $n\in \mathbb N$ such that set  $\left\{k\in\mathbb N : f(n,k)=g(k)\right\}$ has an infinite number of elements?

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Source and history

task_id
fc06abf7-1bfa-51c1-b6b6-53865b2b1190
task_key
test--fc06abf7-1bfa-51c1-b6b6-53865b2b1190
task_revision_id
1
config / split
/ test

Official source

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