Question

3. Let X and Y be two nonempty finite sets. Let F(X, Y) denote the set of all function from X to Y. Is this set finite, countably infinite, or uncountable? Prove your answer

From the class Introduction to Abstract Algebra on the section of countable and uncountable sets

Answer 1

Let the cordiality of X = n and the cordiality of y = m as they are finite n, m elementof N and n, m < infinity now let X = { x_1, x_2,...x_n} then x_1 can go to one of the m elementof y say x_1 rightarrow y: elementof y for this, x_2 can have m choice i.e x_2 rightarrow y: elementof Y and so on. So for each x_i we have m choice. Therefore we will have m times m times ... times m (n times) choices. Implies m^n choices and for each choice we get a forms from X to y. so we will get m^n forms > and any forms f: x rightarrow y is one of that m^n form. We will have a image for each x_i in y. so the number of forms is m^n and on n, m < infinity m^n is finite. Hence