1.7. Proof of Bijection

Now, I am going to make certain claims about this function g. I am going to prove that this function g is a bijection and if it is a bijection then as per the rules of cardinality it shows that the cardinality of F is same as the cardinality of ℤ^r.

In this chunk, we set the premise for proving that the mapping function g from the set of tuples ℤ^r to the finite field F is a bijection. A bijection means that every element in one set is paired with exactly one unique element in the other set, which implies that the two sets have the same size or cardinality.

Think of a bijection like a perfect pairing at a dance. Each dancer (element in one set) has a unique partner (element in the other set). If every dancer has a partner and no partners are left out, the two groups have equal numbers.

Well, proving that g is a surjection is trivial. That comes from the definition of your spanning set. Since as per my definition, the collection of f_1 to f_r is a spanning set.

In this part, we discuss that showing g is a surjection, meaning every element in the target set F must be reachable by some input from ℤ^r, is straightforward. This is established from the definition of a spanning set, which guarantees that any element x in F can be represented as a linear combination of elements from the set f_1 to f_r using appropriate coefficients.

Imagine a chef who can cook any dish using a specific set of spices. If we say the chef's spices span the dishes they can create, then we can be sure that every dish can be made with those spices, just like every field element can be created from the spanning set.

Now, I want to prove that this function g is also an injective function and that I will prove by contradiction.

Here, we are proving that the mapping g is injective, meaning that no two different inputs in ℤ^r yield the same output in F. The proof is attempted through contradiction: if two distinct tuples map to the same element, it would imply that the assumed spanning set is not minimal, which contradicts our initial setup.

Consider trying to identify a suspect in a crime based solely on two distinct witnesses’ accounts. If both witnesses describe a different person, yet you find only one person matching both descriptions, it undermines the credibility of the witnesses. Here, if two tuples lead to the same output, it suggests redundancy in our set, contradicting our assumption of minimalism.

So, that means, whatever I assumed about g is not injective. So that means indeed my mapping g is an injective mapping.

In conclusion, after reaching a contradiction, we confirm that g must be injective, thus supporting that g is a bijection. This critical step shows that the size of our finite field F equals that of ℤ^r. This conclusion reinforces our argument that the number of elements in a finite field is given by the prime power form.

Think of a library where every book has a unique identifier. If two books share the same identifier, it would create confusion about which book you are referencing. Thus, ensuring every book (element) has a unique identifier (mapping) allows the library system to function smoothly, just as our injective function maintains precise one-to-one correspondence.

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