Let's analyze the given functions to determine when they are one-to-one (injective) and onto (surjective) based on the chosen codomain.

One-to-One (Injectivity) Analysis

A function is one-to-one if different inputs (students) map to different outputs. That is, if $f(x_1) = f(x_2)$ implies $x_1 = x_2$.

(a) Mobile Phone Number

• Injectivity: The function is one-to-one if each student has a unique mobile phone number. However, if students share phone numbers (e.g., family members or roommates), then it is not injective.
• Codomain: The set of all possible mobile phone numbers.

(b) Student Identification Number

• Injectivity: This function is typically one-to-one because student ID numbers are uniquely assigned to individuals.
• Codomain: The set of all student identification numbers.

(c) Final Grade in the Class

• Injectivity: This function is not one-to-one because multiple students can receive the same grade.
• Codomain: The set of possible final grades (e.g., {A, B, C, D, F} if letter grades are used).

(d) Hometown

• Injectivity: This function is not one-to-one because multiple students can be from the same hometown.
• Codomain: The set of all possible hometowns.

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Onto (Surjectivity) Analysis

A function is onto if every element in the codomain is mapped to by at least one student.

(a) Mobile Phone Number

• Onto Condition: If the codomain includes only the mobile numbers of students, then the function is onto. If the codomain includes all possible phone numbers (even those not belonging to students), it may not be onto.

(b) Student Identification Number

• Onto Condition: If the codomain is only the set of existing student IDs, then it is onto. If the codomain includes all possible student ID numbers (including unused ones), it is not onto.

(c) Final Grade in the Class

• Onto Condition: The function is onto if every possible final grade (A, B, C, etc.) is assigned to at least one student. If no student receives a certain grade, it is not onto.

(d) Hometown

• Onto Condition: The function is onto if the codomain is restricted to the set of hometowns represented in the class. If the codomain includes all cities in the world, it is likely not onto.

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Summary Table

Function	One-to-One?	Onto Condition
Mobile Phone Number	If each student has a unique phone number	If codomain is only students' phone numbers
Student ID Number	Yes, IDs are unique	If codomain is only student IDs
Final Grade	No, multiple students can have the same grade	If all possible grades are assigned
Hometown	No, multiple students can be from the same place	If codomain is restricted to students' hometowns

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Do you have any questions or need further explanation?

Here are 5 related questions to explore this further:

1. How do we formally define injectivity and surjectivity using mathematical notation?
2. Can a function be both injective and surjective? What is such a function called?
3. How does the choice of codomain affect whether a function is onto?
4. Can we modify the final grade function to make it one-to-one?
5. How would these functions change if applied to a larger student population?

Tip:

A function is bijective if it is both one-to-one and onto. Such functions have unique, invertible mappings between domain and codomain.