Let \(\mathrm{E}=\\{1,2,3,4\\}, \mathrm{F}=\\{\mathrm{a}, \mathrm{b}\\}\) then the number of onto function from \(\mathrm{E}\) to \(\mathrm{F}\) is (a) 14 (b) 16 (c) 12 (d) 32

An onto function (or surjective function) is a function in which every element in the codomain (set \(\mathrm{F}\)) is mapped to by at least one element from the domain (set \(\mathrm{E}\)). If \(f:\mathrm{E}\to\mathrm{F}\) is surjective, then for every element \(y \in \mathrm{F}\), there exists at least one element \(x \in \mathrm{E}\), such that \(f(x)=y\).

In order to find the number of onto functions from \(\mathrm{E}\) to \(\mathrm{F}\), we will apply the Principle of Inclusion-Exclusion (PIE). The PIE is used to find the number of elements in a union of multiple sets, and in this case, it will help us determine the number of onto functions between the two sets. For two sets (in this case, \(\mathrm{F} = \\{ \mathrm{a}, \mathrm{b} \\}\)), the PIE formula is: \(|\mathrm{A}\cup\mathrm{B}| = |\mathrm{A}| + |\mathrm{B}| - |\mathrm{A}\cap\mathrm{B}|\), where \(|\mathrm{A}|\) and \(|\mathrm{B}|\) are the number of elements in set \(\mathrm{A}\) and set \(\mathrm{B}\), respectively.

Using the PIE formula, we can find the number of onto functions from \(\mathrm{E}\) to \(\mathrm{F}\). First, let's consider the number of functions from \(\mathrm{E}\) to \(\mathrm{F}\) without any restrictions. There are 2 elements in set \(\mathrm{F}\) and 4 elements in set \(\mathrm{E}\). Thus, there are a total of \(2^4 = 16\) functions. Now, let's consider the functions that are not onto. There are 2 types of functions that are not onto: 1. Functions in which all elements of set \(\mathrm{E}\) map to \(\mathrm{a}\). There is only one such function. 2. Functions in which all elements of set \(\mathrm{E}\) map to \(\mathrm{b}\). There is only one such function. Therefore, there are a total of \(16 - (1 + 1) = 14\) onto functions from \(\mathrm{E}\) to \(\mathrm{F}\). So, the correct answer is: (a) 14