Talk:Union of Inverses of Mappings is Inverse of Union of Mappings

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Merger with Preimage of Union under Mapping/Family of Sets

Unless I'm mistaken, Preimage of Union under Mapping/Family of Sets is at most a direct corollary to this theorem. Is this what is intended by the merge recommendation? --Keith.U (talk) 12:18, 2 July 2016 (UTC)

Difficult to tell. Links are needed to the concepts invoked. In particular, clarification as to exactly what is meant by the "inverse of a union of a family of mappings" -- inverse of mapping needed, union of family of mappings needed.
Also worth noting that the structuring of this area is suboptimal: the parent page concerns the inverses of two relations, this one is about the inverses of any arbitrary family of mappings. --prime mover (talk) 12:30, 2 July 2016 (UTC)
Would it be wise to create a definition page for the union of functions? When functions are considered as a set of ordered pairs, the definition is clear. However, many readers might be unfamiliar with this concept.
Should already exist. Should be able to find it in Category:Definitions/Set Union if I remember correctly. --prime mover (talk) 10:12, 3 July 2016 (UTC)
I can't seem to find it. Closest I've found is Definition:Union Mapping, though this is not the same. I have some texts that speak of unions of mappings, though none take the time to define it. --Keith.U (talk) 10:38, 3 July 2016 (UTC)
Definition:Union Mapping is indeed what it is. Same will apply to a family of sets, but unless all the mappings agree on all the sets in the family, the concept is of limited use. You will of course find that the union of the sets is a relation,, but then you might as well just rewrite this in the context of relations. And once you have established the need for the functions to agree, you're near enough in the realm of Union of Functions Theorem, which is probably as far as you can go. --prime mover (talk) 10:53, 3 July 2016 (UTC)
So would it be best for me to just reference Definition:Union of Family? The assumption is not made that the functions agree on the intersections of their domains. My reference for formatting is Union of Functions Theorem, though no definition of "union of family of mappings" is given here. I'll let your word be the last here: this has been a lot of discussion over a tiny detail :) --Keith.U (talk) 11:14, 3 July 2016 (UTC)
The fact that no mention of the fact that the mappings agree on the intersections of their domains is not a "tiny detail", it is an important detail -- in that if they do not agree, then their union is not a mapping. Let $A = \{1, 2\}, B = \{a, b\}$. Then let $f_1 = \{(1, a), (2, a)\}, f_2 = \{(1, b), (2, b)\}$. Then $f_1 \cup f_2 = \{(1, a), (2, a), (1, b), (2, b)\}$ and so is not a mapping. So this turns into nothing more interesting than Preimage of Union under Relation/Family of Sets applied to the complete domain and range.
And another noob instruction: please keep to the indentation convention in a discussion -- otherwise it becomes difficult to follow the thread of a conversation. --prime mover (talk) 14:12, 3 July 2016 (UTC)
My phrasing was just an attempt to defuse what I felt was me making too much fuss. In my experience, few (if any) details in mathematics are "tiny" :)
I still feel that this result is an equality of relations, while the theorem linked is only an equality of domains (from my understanding, one result is the equality of a set of ordered pairs, the latter is the equality of a set of singletons). I suppose I am not seeing an important detail; I confess this is not a field in which I am particularly well-versed. You won't hear any more protest from me. --Keith.U (talk) 17:40, 3 July 2016 (UTC)
Yes you're right, there does exist that distinction, I had mentally glossed over it in my mind. Yes, one result concerns the domain, one result concerns the mapping / relation itself. I confess to not getting excited over it because I confess that it's not very exciting. --prime mover (talk) 20:55, 3 July 2016 (UTC)
Thanks for the note on the structuring. I'll restructure shortly. --Keith.U (talk) 09:51, 3 July 2016 (UTC)