Injectivity and Surjectivity

In this video, we dive deep into two important concepts in mathematics: injective and surjective functions. These ideas come up often in algebra, calculus, and higher-level math courses, and understanding them is key to mastering the theory of functions.

We start by explaining what a function is, and how it maps inputs to outputs. Then we focus on injective functions (also known as one-to-one functions). You’ll learn what it means for a function to be injective, how to recognize one, and why injectivity ensures that no two different inputs map to the same output.

Next, we explore surjective functions (also known as onto functions). A surjective function ensures that every possible output value is covered by at least one input value. We’ll discuss how to identify surjectivity, why it matters in both pure and applied math, and how it relates to solving equations.

We also compare injectivity and surjectivity side-by-side so you can clearly see the difference. Using examples, diagrams, and real-world analogies, we make these abstract concepts easier to understand. You’ll learn how these properties connect to topics such as inverse functions, bijective functions, and mathematical proofs.

By the end of this video, you will:

Understand the definitions of injective (one-to-one) and surjective (onto) functions.

Be able to test a function for injectivity and surjectivity.

Recognize the difference between injective, surjective, and bijective functions.

See why these properties are important in higher-level math.

Whether you’re preparing for a math exam, studying for a university course, or simply expanding your mathematical knowledge, this guide to injectivity and surjectivity will give you the clarity you need.

If you enjoy math content like this, remember to check out our other videos on key mathematical concepts, problem-solving techniques, and proof strategies.