Take a mapping a and precompose it with the identity mapping i.
By the definition of the identity mapping the resulting composition is equal to a.
i;a = a
(Here ';' represents forward composition. Mathematicians tend to use backward composition represented by '∘' but I find backward composition awkward and error-prone and so avoid using it.)
Now, if there is another mapping j that is different from i, such that
j;a = a
then the mapping a loses information.
By this I mean that if you are given the value of a(x) you cannot always determine what x was.
To understand this properly you may need to work through a simple example by drawing circles, dots and arrows on a piece of paper.
If there is no such j then mapping a is said to be a monomorphism or injection (the set theoretic term) and it does not lose information.
This specification of the property 'loses information' only involves mapping equality and mapping composition.
It does not involve sets or elements of sets.
An example of a mapping that loses information would be the capitalization of strings of letters.
An example of a mapping that you would not want to lose information would be zip file compression.
If you alter the above specification to use post-composition (a;i = a and a;j = a) instead of pre-composition you get epimorphisms or surjections which capture the idea that a mapping constrains all the values in its codomain.
I like to think of this as the mapping does not return uninitialized values or 'junk' as it is sometimes called.
Take a mapping a and precompose it with the identity mapping i. By the definition of the identity mapping the resulting composition is equal to a.
(Here ';' represents forward composition. Mathematicians tend to use backward composition represented by '∘' but I find backward composition awkward and error-prone and so avoid using it.)Now, if there is another mapping j that is different from i, such that
then the mapping a loses information. By this I mean that if you are given the value of a(x) you cannot always determine what x was. To understand this properly you may need to work through a simple example by drawing circles, dots and arrows on a piece of paper.If there is no such j then mapping a is said to be a monomorphism or injection (the set theoretic term) and it does not lose information.
This specification of the property 'loses information' only involves mapping equality and mapping composition. It does not involve sets or elements of sets.
An example of a mapping that loses information would be the capitalization of strings of letters. An example of a mapping that you would not want to lose information would be zip file compression.
If you alter the above specification to use post-composition (a;i = a and a;j = a) instead of pre-composition you get epimorphisms or surjections which capture the idea that a mapping constrains all the values in its codomain. I like to think of this as the mapping does not return uninitialized values or 'junk' as it is sometimes called.
Bartosz Milewski works through this in more detail (including from the set-theoretic side) in the last 10 minutes of https://www.youtube.com/watch?v=O2lZkr-aAqk&list=PLbgaMIhjbm... and the first 10 minutes of https://www.youtube.com/watch?v=NcT7CGPICzo&list=PLbgaMIhjbm....