Understanding Surjections, Injections, and Bijections

This video tutorial covers the concepts of surjective, injective, and bijective functions. It begins with an introduction to surjective functions, explaining that every element of the codomain must be an image of at least one element from the domain. The video provides examples to illustrate surjective and non-surjective functions. It then introduces injective functions, where each element of the codomain is the image of at most one element from the domain, and provides examples of injective and non-injective functions. Finally, the video explains bijections, which are functions that are both injective and surjective, and compares these concepts side by side.

What is a key characteristic of a surjective function?

• The function has no elements in the codomain.

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

• The function is not defined for any element in the domain.

• Every element of the domain is mapped to a unique element in the codomain.

Which of the following is true for a surjective function?

• It covers all elements in the codomain.

• It maps every element of the domain to a unique element in the codomain.

• It maps every element of the codomain to itself.

• It has no elements in the domain.

How do surjective functions handle elements of the codomain?

• They map every element of the domain to a unique element in the codomain.

• They miss some elements of the codomain.

• They map every element of the codomain to itself.

• They ensure every element of the codomain is mapped by at least one element from the domain.

Why is the function f(n) = 2n not surjective when mapping integers to integers?

• Because it maps integers to non-integers.

• Because it only maps even integers to the codomain.

• Because it maps integers to negative numbers.

• Because it maps every integer to itself.

What defines an injective function?

• 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.

• The function has no elements in the domain.

• The function maps every element of the domain to itself.

What happens to elements of the codomain in an injective function?

• They are missed and not mapped.

• They are the image of multiple elements from the domain.

• They are the image of at most one element from the domain.

• They are repeated as images.

In the context of injective functions, what does 'at most one' mean?

• Exactly one element from the domain maps to each element in the codomain.

• Zero or one element from the domain maps to each element in the codomain.

• Zero or more elements from the domain map to each element in the codomain.

• More than one element from the domain maps to each element in the codomain.

Why is the function g not injective in the given example?

• Because it maps multiple elements of the domain to the same element in the codomain.

• Because it maps every element of the domain to a unique element in the codomain.

• Because it does not map any element of the domain to the codomain.

• Because it maps every element of the codomain to itself.

What is a bijective function?

• A function that is both injective and surjective.

• A function that is neither injective nor surjective.

• A function that is only surjective.

• A function that is only injective.

What is the result of a function being both injective and surjective?

• The function is bijective.

• The function is undefined.

• The function is neither injective nor surjective.

• The function is only surjective.