Functions: Injective, Surjective, and Bijective

The video tutorial explains the concept of functions mapping between sets using two-line notation. It covers the total number of functions possible, their injective properties, and why none are surjective or bijective. The tutorial provides a detailed breakdown of each function's mapping and discusses the conditions for injective, surjective, and bijective functions.

What is the total number of functions that can be formed from a set with elements 1 and 2 to a set with elements a, b, and c?

• 15

• 12

• 9

• 6

In two-line notation, what does it mean if both outputs of a function are the same?

• The function maps both inputs to the same element

• The function is bijective

• The function is surjective

• The function is injective

How many functions are injective when mapping from a set with elements 1 and 2 to a set with elements a, b, and c?

• 9

• 6

• 0

• 3

Which of the following is true for an injective function?

• Every element of the codomain is the image of at most one element from the domain

• The function has more outputs than inputs

• Every element of the codomain is the image of at least one element from the domain

• The function maps all elements to a single element

What is the definition of a surjective function?

• The function has more inputs than outputs

• The function maps all elements to a single element

• Every element of the codomain is the image of at least one element from the domain

• Every element of the codomain is the image of at most one element from the domain

Why are none of the functions in the given set surjective?

• They have more inputs than outputs

• They do not cover all elements of the codomain

• They map all inputs to a single output

• They are all injective

What is a bijective function?

• A function that is both injective and surjective

• A function that maps all inputs to a single output

• A function with more outputs than inputs

• A function that is neither injective nor surjective

Why are there no bijective functions in the given set?

• The number of elements in the domain and codomain are not equal

• The functions map to more than one element

• All functions are surjective

• All functions are injective

What must be true for a function to be bijective?

• The domain must have more elements than the codomain

• The function must be neither injective nor surjective

• The function must map all inputs to a single output

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

Which of the following is not a characteristic of the functions discussed?

• None are surjective

• None are bijective

• They all map from a set with two elements to a set with three elements

• They are all injective