Question

Let \[E = \left\{ {1,2,3,4} \right\}\] and \[F = \left\{ {1,2} \right\}\] then the number of onto function from E to F is
A) 14
B) 16
C) 12
D) 8

Answer 1

Hint:
Here we will use the basic concept of onto function. Onto functions have multiple images in the domain. Then by using the concept of this we will find out the number of onto function from E to F.

Complete step by step solution:
Given sets of data is \[E = \left\{ {1,2,3,4} \right\}\] and \[F = \left\{ {1,2} \right\}\].
We have to find the onto function from E to F, which means that each element of the E can be mapped to any of the two elements of the F. We know that the number of functions from E to F will be equal to the cardinality of the set F raise to the power cardinality of the set E. Therefore, we get
Number of onto functions \[ = {\left| F \right|^{\left| E \right|}}\]
 \[ \Rightarrow \] Number of onto functions \[ = {2^4}\]
Applying the exponent on the term, we get
 \[ \Rightarrow \] Number of onto functions \[ = 16\]
Hence, the number of onto function from E to F is 16.

So, option B is the correct option.

Note:
Here we should note the difference of the one-one and onto. One-one functions are the functions that have their unique image in the domain and onto functions are the functions which have multiple images in the domain. If the function is not one-one then the function is generally known as many one function.
One-one functions are generally known as injective functions and onto functions are generally known as the surjective functions.
Bijective Functions are the functions which are both one-one and onto.