To solve this problem, let's break down the concept of a function being onto, or surjective.

A function $f: X \to Y$ is said to be onto (or surjective) if every element in the co-domain $Y$ is mapped to by at least one element in the domain $X$. In other words, for every $y \in Y$, there exists an $x \in X$ such that $f(x) = y$.

Let's now analyze each statement to see which two are incorrect:

1. "f is onto ⇔ every element in its co-domain is the image of some element in its domain."

   • This is a correct statement. It accurately describes the definition of an onto function.

2. "f is onto ⇔ every element in its domain has a corresponding image in its co-domain."

   • This is incorrect. This describes a basic property of functions (every element in the domain has an image) but not the property of being onto. For a function to be onto, the focus is on covering all elements of the co-domain, not just the existence of images.

3. "f is onto ⇔ ∀y ∈ Y, ∃x ∈ X such that f(x) = y."

   • This is correct. It expresses the idea that for every element in the co-domain $Y$, there exists an element in the domain $X$ such that $f(x) = y$. This is the formal definition of an onto function.

4. "f is onto ⇔ ∀x ∈ X, ∃y ∈ Y such that f(x) = y."

   • This is incorrect. This statement describes a basic property of functions (that every $x \in X$ is mapped to some $y \in Y$) but does not imply that all elements in $Y$ are covered. This doesn't guarantee that the function is onto.

5. "f is onto ⇔ the range of f is the same as the co-domain of f."

   • This is correct. For a function to be onto, its range (the set of all outputs) must equal the co-domain.

Summary:

The two incorrect statements are:

• Statement 2: "f is onto ⇔ every element in its domain has a corresponding image in its co-domain."
• Statement 4: "f is onto ⇔ ∀x ∈ X, ∃y ∈ Y such that f(x) = y."

Would you like further clarification on any part of this?

---

Here are 5 related questions to expand your understanding:

1. What is the difference between an onto function and a one-to-one (injective) function?
2. How do you prove that a function is not onto?
3. Can a function be both injective and surjective? What is the term for such functions?
4. How does the concept of onto apply to functions between finite sets?
5. What is the difference between the domain, range, and co-domain of a function?

Tip: To test whether a function is onto, focus on checking if every element in the co-domain has a pre-image in the domain.