Final answer:
To establish that m=n for a bijection f from [m] to [n], we use mathematical induction, demonstrating that for each element in [m] there is a unique element in [n] and that this correspondence holds when incrementing m and n by 1.
Step-by-step explanation:
To prove that m=n when f is a bijection from [m] to [n], one can use induction. First, let's define [m] as the set {1, 2, ..., m} and similarly, [n] as the set {1, 2, ..., n}. A bijection is a function that is both injective (one-to-one) and surjective (onto). Therefore, each element in [m] must have a unique partner in [n] and vice versa.
Starting with the base case, if [m] is a non-empty set, then there must be at least one element in [n] since f is surjective. Let's take m = 1. There has to be exactly one element, say n, in the co-domain such that f is still bijective. Hence, for m=1, n must also be 1.
For the inductive step, presume that for a set [k] where k<=m, n=k. Now, consider [m+1]. If f is still a bijection over [m+1], there must be exactly one additional unique element in [n], implying that the co-domain must now be [n+1]. This shows that for f to remain a bijection, m+1 must be equal to n+1, maintaining the property that m=n.
By induction, we have shown that for f to be a bijection, the size of the domain ([m]) must equal the size of the co-domain ([n]), which means m=n. An example to illustrate this could be if m=2, and assuming f is bijective, then n must also be 2; this can be demonstrated by defining a bijective function f such that f(1)=1 and f(2)=2.