Ménage à trois
Counting maps, Discrete components, and Relation (Equation)
It takes [a saying says] only 2 Indians to form 3 political parties. I venture, emboldened by this sentence, into a collection (which, in the hope of saving some space, is christened set) A of two elements. Let’s name, so as to establish that what we have here is a crackhead, both elements (of the set A) a i.e. A = {a, a}.
I like pictures (notwithstanding the fact that pictures fail to adequately capture my handsomeness); so we take a picture—veridical picture–of the set A in the set A. There may very well be many ifs and buts, but veridical–as is–is one of a kind, which is my way of saying that we are looking at the identity function 1A: A -> A. I don’t (so does everybody else for that matter) need to explain taking a picture of a picture.
How about pictures of 1A in 1A? There is, for hors d’œuvre (photocopying bits and pieces of the above), a picture 11A: 1A -> 1A mapping the two discrete components (advance apologies for I’m not certain of being true to the definitions [Conceptual Mathematics, page 358 – ]; thank God for tomorrow) of the domain object 1A (of the picture 11A) to two different discrete components of the codomain object 1A. Then there is another picture f: 1A -> 1A (a map of sets, just like 11A, satisfying f1A = 1Af; note that now we are in the category of endomaps [Conceptual Mathematics, page 136]) mapping both discrete components of the domain object 1A to one of the two discrete components of the codomain object 1A–but which one?
Ah, now I see why people give different names to different people, things, and places. We have, with different names as in A = {a1, a2}, a description of the dynamical system 1A: A -> A in terms of generators and relations (Generators: a1, a2; Relations: 1A(a1) = a1, 1A(a2) = a2). The description, with the same name as in A = {a, a}, of our automaton 1A: A -> A looks leaner (Generator: a; Relation: 1A(a) = a). Or, does it really?
What does, if it does and if it doesn’t, this Generator / Relation side-business have to do with our main business of counting maps? Does it have anything to do with components (discrete or otherwise, which, by the way, you sneaked in surreptitiously) at least? Can I show / deduce / derive / prove that there are two different pictures of 1A in 1A when both elements of the set A have the same name as in A = {a, a}, and that there are four pictures of 1A in 1A in the case of A = {a1, a2} if I learn all about describing whatever it is that we are describing (Conceptual Mathematics; pages 161 – , and 182 – )? Bribe me (I’m no Gita-thumpin doer–the point of deeds, for me, is fruits 😉
Manana…
Conceptual Mathematics
Conceptual Mathematics 
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