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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Thu, 10 Jul 2025 11:58:59 +0200
Organization: A noiseless patient Spider
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In-Reply-To: <104o27c$pvdh$1@dont-email.me>

On 10.07.2025 11:47, Mikko wrote:
> On 2025-07-09 16:35:02 +0000, WM said:
> 
>> On 09.07.2025 10:34, Mikko wrote:
>>> On 2025-07-08 15:47:12 +0000, WM said:
>>>
>>>> On 08.07.2025 09:46, Mikko wrote:
>>>>> On 2025-07-07 15:37:08 +0000, WM said:
>>>>>
>>>>>> On 07.07.2025 10:29, Mikko wrote:
>>>>>>
>>>>>>>>>> Bijection requires completeness of domain and codomain.
>>>>>>>>>
>>>>>>>>> So you say but cannot prove.
>>>>>>>>
>>>>>>>> It is so by definition. See e.g. W. Mückenheim: "Mathematik für 
>>>>>>>> die ersten Semester", 4th ed., De Gruyter, Berlin (2015).
>>>>>>>
>>>>>>> Can you refer to some better author?
>>>>>>
>>>>>> That is hardly feasible. But you can look up the definition in 
>>>>>> every textbook of your choice. You will find the same result. Even 
>>>>>> Wikipedia will be sufficient: a bijection is a relation between 
>>>>>> two sets such that each element of either set is paired with 
>>>>>> exactly one element of the other set.
>>>>>
>>>>> So no requirement of completeness.
>>>>
>>>> "Each element" means that none is missing.
>>>
>>> No, it does not. What is said about each element applies to missing
>>> elements, too.
>>>
>> In mathematics, a surjective function (also known as surjection, or 
>> onto function is a function f such that, for every element y of the 
>> function's codomain, ... [Wiki]
> 
> That text does not say that
>> There is none missing.

Try to learn to understand mathematical texts. Every means none is 
missing, formally abbreviated by the universal quantifier ∀. Until you 
got it: EOD.

Regards, WM