2 - Russell’s Paradox. Cardinality. Uncountability. Integers and modular arithmetic

Question 1

What does injective mean?

Answer 1

An injective function is a function that maps distinct elements of its domain to distinct elements of its codomain.

In other words, every element of the function’s codomain is the image of at most one element of its domain.

Injective means we won’t have two or more domain element “A”s pointing to the same domain element “B”.

We can have a “B” without a matching “A”

Injective is also called “One-to-One”

Question 2

What does surjective mean?

Answer 2

A surjective function is a function where every element of the codomain is mapped to an element of the domain. A codomain element can have more than one domain element mapped to it.

In other words, surjective means that every codomain “B” has at least one matching domain “A” (maybe more than one).

There won’t be a “B” left out.

Question 3

What is bijective?

Answer 3

A bijective function is a function that is both injective and subjective together.

Think of it as a “perfect pairing” between the sets: everyone has a partner and no one is left out.

So there is a perfect “one-to-one correspondence” between the members of the sets.