Functions: Injective, Surjective, and Bijective

The video tutorial explains how to map functions from a set with elements 1, 2, and 3 to a set with elements a and b using two-line notation. It identifies eight possible functions and examines their properties, such as injectivity, surjectivity, and bijectivity. The tutorial concludes that none of the functions are injective, six are surjective, and none are bijective, highlighting the conditions required for each type of function.

How many possible functions can be created when mapping a set of three elements to a set of two elements?

• 6

• 10

• 4

• 8

In two-line notation, what do the elements in the first row represent?

• Codomain

• Inputs

• Range

• Outputs

What is the main characteristic of an injective function?

• Every element of the domain maps to a unique element in the codomain

• Every element of the codomain is mapped by at least one element of the domain

• The function is both injective and surjective

• The number of elements in the domain and codomain are equal

Why are none of the functions in this scenario injective?

• All elements of the codomain are mapped by multiple elements of the domain

• The domain and codomain have different numbers of elements

• The functions are not surjective

• The functions are not bijective

What defines a surjective function?

• Every element of the domain maps to a unique element in the codomain

• Every element of the codomain is mapped by at least one element of the domain

• The function is both injective and surjective

• The domain and codomain have the same number of elements

Which functions in this scenario are surjective?

• f1 and f2

• f3 through f8

• All functions

• None of the functions

How many functions are neither injective nor surjective?

• 0

• 2

• 4

• 6

What is a necessary condition for a function to be bijective?

• The function must be neither injective nor surjective

• The domain and codomain must have the same number of elements

• The function must be surjective

• The function must be injective

How many bijective functions are there in this scenario?

• 6

• 4

• 2

• 0

Why can't any of the functions in this scenario be bijective?

• The functions are not injective

• All of the above

• The functions are not surjective

• The domain and codomain have different numbers of elements