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\[f\colon\Bbb Z\to\{28,29\}~,~ f(x)= \begin{cases} 29 \ \ \text{if} \ [x] \in \{[4k]\mid 0\leq k\leq 99~\land~k\notin\{25,50,75\}\} \ \\ 28 \ \text{otherwise} \end{cases} \]

Find a function \(g\colon\Bbb Z^+\cup\{0\}\to\{28,29\}\) which is not piece-wise defined and is identical to \(f\) in its own domain.

The function you should be seeking for might not be that mathematical....


Clarifications:

  • \([x]\) denotes the congruence class of \(x\) modulo \(400\).
This problem is original

Note by Nihar Mahajan
10 months ago

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Wha do you mean by not piecewise defined? Agnishom Chattopadhyay · 10 months ago

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@Agnishom Chattopadhyay This should be helpful. Prasun Biswas · 10 months ago

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@Prasun Biswas I know what piecewise means. What is an example of a function not piecewise? Agnishom Chattopadhyay · 10 months ago

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@Agnishom Chattopadhyay Something like \(f(x)=x^2\), no? I don't see a formal definition of non-piecewise anywhere, so I guess there's a scope for ambiguity. I can't do a better phrasing for a troll (not quite mathematical) problem. Prasun Biswas · 10 months ago

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@Prasun Biswas Haha, a better phrasing could be an elementary function, a function with a closed form, etc. Agnishom Chattopadhyay · 10 months ago

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Well , do you guys want me to reveal the answer? Nihar Mahajan · 10 months ago

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@Nihar Mahajan I think the question you're really trying to as is "What is f better known as"?

The answer to that is f(x) is the number of days in the february of year x Agnishom Chattopadhyay · 10 months ago

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@Agnishom Chattopadhyay Correct..... Nihar Mahajan · 10 months ago

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I do not understand the problem. We already have defined f in the problem statement. How can we improve upon that? Agnishom Chattopadhyay · 10 months ago

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@Agnishom Chattopadhyay You have to find all f(x) which have those two properties. Nihar Mahajan · 10 months ago

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@Nihar Mahajan I think you mean a function \(f(x)\) which satisfies those two properties but is not piecewise defined?

The problem, as is currently phrased, doesn't make sense since we already have that \(f(x)\), piecewise defined! You don't find stuff that suits a definition, you define stuff and go from there. Prasun Biswas · 10 months ago

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@Prasun Biswas I realized that the rephrased statement is even more silly , I have again rephrased lol Nihar Mahajan · 10 months ago

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@Prasun Biswas I have rephrased the problem statement. I hope it does not cause any issue now. Nihar Mahajan · 10 months ago

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@Prasun Biswas I am not much familiar with "piece-wise" defined. But for instance , a function say \(f(x)=x^2\) satisfies those properties.(It does not though) Nihar Mahajan · 10 months ago

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