tl;dr: The cardinality of power set of natural numbers is the continuum (c).

Background information

Power set: Given a set S, the power set of S is the set of all subsets of S. The order of a power set of a set of order n is 2^n. Power sets are larger than the sets associated with them. The power set of S is variously denoted 2^S or (script capital p)(S).

The power set of a given set s can be found in Mathematica using Subsets[s].

Injective: tl;dr: Injective means one to one.

Let f be a function defined on a set A and taking values in a set B. Then f is said to be an injection (or injective map, or embedding) if, whenever f(x) = f(y), it must be the case that x = y. Equivalently, x!=y implies f(x)!=f(y). In other words, f is an injection if it maps distinct objects to distinct objects. An injection is sometimes also called one-to-one.

A linear transformation is injective if the kernel of the function is zero, i.e., a function f(x) is injective iff Ker(f) = 0.

A function which is both an injection and a surjection is said to be a bijection.

In the categories of sets, groups, modules, etc., a monomorphism is the same as an injection, and is used synonymously with (open curly double quote)injection(close curly double quote) outside of category theory.

(to be continued)

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