Cantor's Theorem

If there is a bijection between two sets, they have the same number of elements (are equinumerous, or have the same cardinality). Arrangements: A mapping, also known as a function, is a rule that associates elements from one set with elements from another. This is how we write it:  f : X → Y , f is referred to as the function/mapping, the set X is referred to as the domain, and Y is referred to as the codomain. We specify the rule by writing f(x) =y or f : x 7→ y. e.g. X = {1, 2, 3}, Y = {2, 4, 6}, the map f(x) = 2x associates each element x ∈ X with the element in Y which means to double it.

In this article we are going to discuss cantor's intersection theorem, state and prove cantor's theorem, cantor's theorem proof.

A bijection is a mapping that is injective as well as surjective.

• Injective (one-to-one): A function is injective if it takes each element of the domain and applies it to no more than one element of the codomain. It never maps more than one domain element to the same codomain element. Formally, if f is known to be a function between namely set X as well as set Y , then f is injective iff ∀a, b ∈ X, f(a) = f(b) → a = b 

• Surjective (onto): If a function maps something onto every element of the codomain, it is surjective. It can map multiple things to the same element in the codomain, but it must hit every element in the codomain.Formally, if f is known to be a function between set X and set Y , then f is surjective (if and only if)  iff ∀y ∈ Y, ∃x ∈ X, f(x) = y 

The Heine-Cantor theorem The cardinality of any set A is strictly less than the cardinality of A's power set : |A| < |P(A)| 

Proof: To prove this, we will show (1) that |A| ≤ |P(A)| and then (2) that ¬(|A| = |P(A)|). This is equivalent to the strictly less than phrasing in the statement of the given theorem. (1) |A| ≤ |P(A)| : Now , to show this, we just need to produce a bijection between A as well as a subset of P(A). Then we'll know A is the same size as that subset, which cannot be larger than P. (A).

Consider the set E = {{x} : x ∈ A}, the set of all single-element subsets of A. Clearly E ⊂ P(A), because it is made up of various subsets of A. Incidentally, it is a proper subset, since we know it doesn’t contain ∅. 

The map g : A → E defined by g(x) = {x} is one-to-one and onto. How do we know this? (This is laboured, but useful to be certain that you understand this!) 

• One-to-one: Let’s suppose we have x, y ∈ A and g(x) = g(y). Then by the definition of injectiveness above, we want to be sure that this means x = y, if g is going to be one-to-one. g(x) = {x} and g(y) = {y}, so, g(x) = g(y) means {x} = {y}. These two one-element sets can only be equal if their members are equal, so x = y. Therefore g is one-to-one.

• Onto: Is it true that ∀y ∈ E, ∃x ∈ A, g(x) = y? Yes. We know that E = {{x} : x ∈ A} so ∀y ∈ E, ∃x ∈ A such that y = {x}. And that is because each element of E just is a set with an element from A as its sole member. And since g(x) = {x}, we have ∀y ∈ E, ∃x ∈ A, g(x) = y, so g is surjective.

Therefore |A| = |E| ≤ |P(A)| . 

Importance of Cantor's Theorem

Cantor's theorem had immediate and significant implications for mathematics philosophy. For example, taking the power set of an infinite set iteratively and applying Cantor's theorem yields an infinite hierarchy of infinite cardinals, each strictly larger than the one before it.

Cantor was successful in demonstrating that the cardinality of the power set is strictly greater than that of the set for all sets, including infinite sets. (In fact, the cardinality of the Reals is the same as the cardinality of the Integers' power set.) As a result, the power set of the Reals is larger than that of the reals.