Answer
The two **incorrect** statements are:
**(b)** and **(d).**
Work Step by Step
First, recall that a function \(f : X \to Y\) is **onto** (or **surjective**) precisely when:
> For **every** \(y \in Y\), there exists **at least one** \(x \in X\) such that \(f(x) = y.\)
Equivalently,
> The **range** (or image) of \(f\) is exactly the entire codomain \(Y\).
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## Analyzing Each Statement
1. **(a)** *“\(f\) is onto \(\iff\) every element in its co-domain is the image of some element in its domain.”*
- This is the **correct** definition of onto.
- **Conclusion:** **Correct**.
2. **(b)** *“\(f\) is onto \(\iff\) every element in its domain has a corresponding image in its co-domain.”*
- **Every** function \(f: X \to Y\) does this by definition—each \(x\in X\) has **some** \(f(x)\in Y\). It does **not** capture onto-ness.
- **Conclusion:** **Incorrect** (because it describes something true of all functions, not specifically onto functions).
3. **(c)** *“\(f\) is onto \(\iff\) \(\forall y\in Y\), \(\exists x\in X\) such that \(f(x)=y\).”*
- This is precisely the standard logical definition of onto.
- **Conclusion:** **Correct**.
4. **(d)** *“\(f\) is onto \(\iff\) \(\forall x\in X\), \(\exists y\in Y\) such that \(f(x)=y\).”*
- Again, this only states that \(f\) is a function (every \(x\) maps to some \(y\)), which is **always** true. It says nothing about hitting **all** \(y \in Y\).
- **Conclusion:** **Incorrect**.
5. **(e)** *“\(f\) is onto \(\iff\) the range of \(f\) is the same as the co-domain of \(f\).”*
- This is the standard alternative definition of onto (surjective).
- **Conclusion:** **Correct**.
