Genady

I suppose it was not serious.

It is often mentioned that the number theory has applications in computer science, esp. in cryptography. It seems to me that these applications are very few compared to the size of the number theory itself, aren't they? More generally, are there any applications of the number theory that are not in computer science?

Here are two of them mating, but the picture didn't turn out clear.

Yeah. And I can quote Mao Zedong stuff and compare it to quantum machines. Still BS.

BS

Check this exercise: I think, the text is mistaken. In the category of sets, a direct product of A and B is their Cartesian product C, and the morphisms are maps from C to A and from C to B, which are not injective, i.e., monomorphisms, but rather surjective, i.e., epimorphisms. OTOH, a direct sum of sets A and B is their disjoint union C, and the morphisms are maps from A to C and from B to C, which are not surjective (epimorphisms), but rather injective (monomorphisms). Do you agree?

Today I've found a simple example of a morphism which is monomorphism, i.e., one-to-one, and epimorphism, i.e., onto, but is not an isomorphism, i.e., does not have an inverse.

Here it is: Mapping [math]f: A \to B[/math] is onto if for any X and any mappings [math]p: B \to X[/math] and [math]q: B \to X[/math], [math]p \circ f = q \circ f \Rightarrow p=q[/math].

TIL that "one-to-one" mapping between two sets can be defined as an external property of the mapping, i.e., without any reference to elements of the sets and to what happens to them under the mapping. Here we go: The mapping [math]f: A \to B[/math] is one-to-one if for any X and any mappings [math]p: X \to A[/math] and [math]q: X \to A[/math], [math]f \circ p = f \circ q \Rightarrow p=q[/math]. Can you come up with a similarly external definition of "onto" mapping?

It's from fb: https://www.facebook.com/permalink.php?story_fbid=pfbid028dPTEhRvxf6j3HMNy52Ymv742GMuQXekebvDo8DzjkAU9EuiENkDdM1Kt2TAUM8hl&id=100090706310719&__cft__[0]=AZZHIbs76gT7OBtsHj6mMXIx5moXtwff6b04aSxChSwB5lEXYpPMSGOGKRrOS9uci6M4uNynPdHRcXUqgAn7915tu-8VZC_TjOrw2r4iP4O92OMOlQ6FBa5PGInZyi69m869_BZmfljXXH2ce7DMOVWjbE4olX7ZzZg8C7glt3qkEg&__tn__=%2CO%2CP-R

These are not words in English.

Yes, it does. It is an effective manipulation technique.

Because it works.

I remember getting a question on my US citizenship exam (many-many years ago), "What is Constitution?" with one line for an answer. I've answered, correctly, "Constitution is the supreme law of the land."