How to Prove a Function is Not Surjective(Onto)

TL;DR

Explaining the concept of surjective functions by proving the function is not subjective using a simple example.

Transcript

being asked to prove that a certain function is not subjective so in order to do a problem like this you really have to understand what it means for a function to be surjective so F from capital x to Capital y is surjective if for all of the Y values in the co-domain capital Y we can find some x value in the domain capital x such that F takes Littl... Read More

Key Insights

• ☕ Surjective functions ensure every element in the co-domain has a pre-image in the domain.
• 🥶 Proving that a function is not surjective involves finding a single counterexample in the co-domain.
• 🚱 Demonstrating non-surjectiveness through a specific example simplifies the proof process.
• ❓ Contradiction can be used to further strengthen the proof of a function not being surjective.

Questions & Answers

Q: What does it mean for a function to be surjective?

A function is surjective if for all Y values in the co-domain, there exists an X in the domain where F(X) maps to Y.

Q: How do you prove a function is not surjective?

To show a function is not surjective, find a Y value for which F(X) does not equal Y, demonstrating an absence of mapping for all X values.

Q: Why is the example 3x + 2 used to prove non-surjectiveness?

The example of 3x + 2 simplifies the proof by showing that there exists a Y value (0) where F(X) is not equal to Y, proving non-surjectiveness.

Q: Can a function be both surjective and non-surjective?

No, a function cannot be both surjective and non-surjective as these properties are mutually exclusive in function theory.

Summary & Key Takeaways

• A surjective function F from X to Y means that for all Y values in Y, there exists an X in X such that F(X) = Y.

• To show F is not surjective, negate the quantifiers and find a Y value where F(X) does not equal Y.

• Using the example 3x + 2, prove that there exists a Y value (0) for which F(X) is not equal to Y, thus showing F is not subjective.