Answer :
To determine which statements about a function being "onto" (also known as surjective) are correct and which are not, let's go through each option and understand the definition of an "onto" function.
A function [tex]\( f: X \to Y \)[/tex] is called onto or surjective if every element in the co-domain [tex]\( Y \)[/tex] is mapped to by at least one element in the domain [tex]\( X \)[/tex].
Now, let's analyze each given statement:
1. Statement 1: "f is onto ⇔ every element in its co-domain is the image of some element in its domain."
- This statement is correct. For a function to be onto, every element of the co-domain [tex]\( Y \)[/tex] must have a pre-image in the domain [tex]\( X \)[/tex].
2. Statement 2: "f is onto ⇔ every element in its domain has a corresponding image in its co-domain."
- This statement is incorrect. While it is true that every element in the domain must map to an element in the co-domain (this is simply the definition of a function), this does not ensure that every element in the co-domain is covered. Hence, this does not imply onto.
3. Statement 3: "f is onto ⇔ ∀ y ∈ Y, ∃ x ∈ X such that f(x) = y."
- This statement is correct. It directly states the condition for a function to be onto: for every element [tex]\( y \)[/tex] in the co-domain, there must exist at least one element [tex]\( x \)[/tex] in the domain such that [tex]\( f(x) = y \)[/tex].
4. Statement 4: "f is onto ⇔ ∀ x ∈ X, ∃ y ∈ Y such that f(x) = y."
- This statement is incorrect. While it states a basic property of functions (every [tex]\( x \)[/tex] must map to some [tex]\( y \)[/tex]), it does not capture the requirement that every [tex]\( y \)[/tex] in the co-domain is mapped to by some [tex]\( x \)[/tex], which is necessary for being onto.
5. Statement 5: "f is onto ⇔ the range of f is the same as the co-domain of f."
- This statement is correct. For a function to be onto, the range (the set of all actual outputs of the function) must cover the entire co-domain.
Thus, the two statements that are incorrect ways to express a function being onto are:
- "f is onto ⇔ every element in its domain has a corresponding image in its co-domain."
- "f is onto ⇔ ∀ x ∈ X, ∃ y ∈ Y such that f(x) = y."
A function [tex]\( f: X \to Y \)[/tex] is called onto or surjective if every element in the co-domain [tex]\( Y \)[/tex] is mapped to by at least one element in the domain [tex]\( X \)[/tex].
Now, let's analyze each given statement:
1. Statement 1: "f is onto ⇔ every element in its co-domain is the image of some element in its domain."
- This statement is correct. For a function to be onto, every element of the co-domain [tex]\( Y \)[/tex] must have a pre-image in the domain [tex]\( X \)[/tex].
2. Statement 2: "f is onto ⇔ every element in its domain has a corresponding image in its co-domain."
- This statement is incorrect. While it is true that every element in the domain must map to an element in the co-domain (this is simply the definition of a function), this does not ensure that every element in the co-domain is covered. Hence, this does not imply onto.
3. Statement 3: "f is onto ⇔ ∀ y ∈ Y, ∃ x ∈ X such that f(x) = y."
- This statement is correct. It directly states the condition for a function to be onto: for every element [tex]\( y \)[/tex] in the co-domain, there must exist at least one element [tex]\( x \)[/tex] in the domain such that [tex]\( f(x) = y \)[/tex].
4. Statement 4: "f is onto ⇔ ∀ x ∈ X, ∃ y ∈ Y such that f(x) = y."
- This statement is incorrect. While it states a basic property of functions (every [tex]\( x \)[/tex] must map to some [tex]\( y \)[/tex]), it does not capture the requirement that every [tex]\( y \)[/tex] in the co-domain is mapped to by some [tex]\( x \)[/tex], which is necessary for being onto.
5. Statement 5: "f is onto ⇔ the range of f is the same as the co-domain of f."
- This statement is correct. For a function to be onto, the range (the set of all actual outputs of the function) must cover the entire co-domain.
Thus, the two statements that are incorrect ways to express a function being onto are:
- "f is onto ⇔ every element in its domain has a corresponding image in its co-domain."
- "f is onto ⇔ ∀ x ∈ X, ∃ y ∈ Y such that f(x) = y."
