What Are Injective and Surjective Functions?

TL;DR

Injective functions map unique elements from the domain to each element in the co-domain, meaning no two domain elements map to the same co-domain element. Surjective functions ensure that every element of the co-domain is mapped to by at least one element in the domain. A function can be both injective and surjective, achieving a one-to-one correspondence between the two sets.

Transcript

In this video I want to introduce you to some terminology that will be useful in our discussion of functions and invertibility. And this is, in general, terminology that you'll probably see in your mathematical careers. So let's say I have a function f, and it is a mapping from the set x to the set y. We've drawn this diagram many times, but it nev... Read More

Key Insights

• 🥶 Functions can be described as mappings between a domain and a co-domain.
• 🥶 A surjective function maps every element in the co-domain to at least one element in the domain.
• 🥶 An injective function ensures that each element in the co-domain is mapped to by at most one element in the domain.
• 🥶 Surjective functions have their image or range equal to the co-domain.
• 🥶 Injective functions have a one-to-one correspondence between elements in the domain and co-domain.
• 🥶 Functions can be both surjective and injective, ensuring that every element in the co-domain is mapped to by exactly one element in the domain, and every element in the co-domain is mapped to.
• 🥶 Not all functions are surjective or injective, and they can have varying degrees of correspondence between the domain and co-domain.

Questions & Answers

Q: What is the difference between a surjective and an injective function?

A surjective function maps every element in the co-domain to at least one element in the domain, while an injective function ensures that each element in the co-domain is mapped to by at most one element in the domain.

Q: Can a function be both surjective and injective?

Yes, a function can be both surjective and injective. This means that every element in the co-domain is mapped to by exactly one element in the domain, and every element in the co-domain is mapped to.

Q: What happens when a function is not surjective?

When a function is not surjective, there are elements in the co-domain that are not mapped to by any element in the domain. This means that the image or range of the function is smaller than the co-domain.

Q: How can we make a function both surjective and injective?

To make a function both surjective and injective, we need to ensure that every element in the co-domain is mapped to by exactly one element in the domain, and every element in the co-domain is mapped to. This can be achieved by adjusting the mappings between elements.

Summary & Key Takeaways

• Functions can be described as mappings from a domain to a co-domain.

• A function is considered surjective if every element in the co-domain has at least one corresponding element in the domain that maps to it.

• An injective function ensures that each element in the co-domain is mapped to by at most one element in the domain.