Question

This question concerns functions \(f:\\{A, B, C, D, E, F, G\\} \rightarrow\\{1,2\\} .\) How many such functions are there? How many of these functions are injective? How many are surjective? How many are bijective?

Short answer

The total number of functions is 128, the number of injective functions is 0, the number of surjective functions is 2, and the number of bijective functions is also 0.

Step-by-step solution

Finding the Total Number of Functions

The total number of functions from a 7-element set (A, B, C, D, E, F, G) to a 2-element set (1, 2) can be calculated using the formula \(m^n\), where m is the number of elements in the range and n is the number of elements in the domain. So the total number of functions is \(2^7 = 128\).

Finding the Number of Injective Functions

An injective function from a 7-element set to a 2-element set is not possible, because to be injective, every element of the domain must map to a unique element of the range. Since the range only has 2 elements, there cannot be an injective function from a set with more than 2 elements. So, the number of injective functions is 0.

Finding the Number of Surjective Functions

For a function to be surjective, every element in the range must be mapped to by at least one element in the domain. The number of surjective functions can be calculated using the formula \((2^7) - 2*(2^6) + 1 = 2\). The formula counts all the functions, subtracts out those that miss one element in the range, and then adds back the one function that misses both elements.

Finding the Number of Bijective Functions

A bijective function from a 7-element set to a 2-element set is not possible because a bijective function is both injective and surjective. Since there are no injective functions from a 7-element set to a 2-element set, there cannot be any bijective functions. So, the number of bijective functions is 0.