Special Kinds of Morphisms

We classify morphisms according to their properties as functions. If Φ : A → A' is a morphism, then we call Φ an isomorphism if Φ is bijective; an epimorphism if Φ is surjective, and a monomorphism if Φ is injective.
   Some further characterisations: The abstract properties of an algebraic system are exactly those whch are invariant (i.e., which do not change) under isomorphism. For epimorphisms, A' is called the homomorphic image of A, and we regard (A', Ω') as an abstraction or a model of (A, Ω). A monomorphism A → A' is sometimes called an embedding of A into A'.
   We single out morphisms that map algebras onto themselves. We called a morphism Φ : A → A' that maps (A, Ω) onto itself an endomorphism. If Φ is also bijective, hence an isomorphism, Φ : A → A, then we call it an automorphism.

ignore this thread

NO EXCEPTIONS, but sage

bmup

>>4
OKAY YOU FUQIN ANGERED AN EXPERT PROGRAMMER
GODFUCKIGNDAMN
FIRST OF ALL, YOU DONT FUQIN KNOW WHAT A MAN PAGE IS
SECONDLY, THIS IS /prog/ DO NOT DEMAND USEFUL ANSWERS THE WAY YOU WANT THEM TO BE
THIRDLY PROGRAMMING IS ALL ABOUT PHILOSOPHY AND ``ABSTRACT BULLSHITE'' THAT YOU WILL NEVER COMPREHEND
AND FUQIN LASTLY, FUCK OFF WITH YOUR BULLSHYT
EVERYTHING HAS ALREADY BEEN ANSWERED IN >>3,4,10
:)

while(8===>/0){

cout << cocks;

}

Nice indentation, cretin.

>>6
This is why we have the forced indentation of code.

[b][i]OKAY YOU FUQIN ANGERED AN EXPERT PROGRAMMER
GODFUCKIGNDAMN
FIRST OF ALL, YOU DONT FUQIN KNOW WHAT A MAN PAGE IS
SECONDLY, THIS IS /prog/ DO NOT DEMAND USEFUL ANSWERS THE WAY YOU WANT THEM TO BE
THIRDLY PROGRAMMING IS ALL ABOUT PHILOSOPHY AND ``ABSTRACT BULLSHITE'' THAT YOU WILL NEVER COMPREHEND
AND FUQIN LASTLY, FUCK OFF WITH YOUR BULLSHYT
EVERYTHING HAS ALREADY BEEN ANSWERED IN >>3,4,10

>>4
see >>5

<-- check em dubz

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Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.

Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well.