I was searching about some methods commomly used in Combinatorics, and I came by the term Bijective functions.
Thus, I searched about it and found out that the fuctions that are Injective and Surjective both are bijective, but I am still confused that how can we apply them in our question ?
Please refer to this content :
Please explain me the meaning of the content and I would prefer if you suggest some ways to implement Bijection in combinatorics.
In combinatorics, we frequently use bijections to count the elements of a set whose size is unknown. For example, suppose we have a well-understood set, say X, and we know the number of elements of X. If we can form a bijection to another set, say Y, we know that the number of elements in Y is the same as the number of elements of X. This is known as a one-to-one correspondence and is referenced as the correspondence principle in your images.
When showing the number of distinct subsets of X = {set with n elements} is $2^n$, the text forms a bijection from a set we know how to count, i.e. the set of all sequences of length n using the symbols 0 and 1 to P (the set we wish to count). By successfully constructing a bijection, we show that the number of elements of P is equal to the number of elements of the set that we can already count.
This approach is useful when directly counting elements of a set are difficult or when studying numbers that count multiple things. (Check out the Catalan numbers for an example).