To determine when the functions mentioned in your problem are one-to-one (also called injective), we need to understand that a function is one-to-one if, for every pair of distinct elements in the domain, their images under the function are distinct as well. In other words, if $f(x_1) = f(x_2)$, then $x_1 = x_2$.

Let's look at each case:

a) Mobile phone number

• Condition for being one-to-one: The function is one-to-one if every student has a unique mobile phone number.
• Explanation: If two different students have the same mobile phone number, the function would not be injective, as different students would be mapped to the same value (the phone number). Therefore, the function is one-to-one only if each student has a different mobile phone number.

b) Student identification number

• Condition for being one-to-one: The function is one-to-one if each student is assigned a unique student identification number.
• Explanation: A student ID number is generally unique for each student. Therefore, under normal circumstances, this function is always one-to-one because no two students will share the same ID number.

c) Final grade in the class

• Condition for being one-to-one: The function is one-to-one if no two students receive the same final grade.
• Explanation: If two students receive the same final grade, the function would not be injective, as it would map both students to the same grade. This function is one-to-one only if every student has a distinct final grade in the class, which is often not the case since multiple students may have the same grade.

d) Home town

• Condition for being one-to-one: The function is one-to-one if every student comes from a unique home town.
• Explanation: If two students are from the same home town, the function would not be injective, as both students would be mapped to the same home town. This function is one-to-one only if each student comes from a different home town, but this is unlikely in most cases, as multiple students may share the same home town.

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Summary of Conditions for One-to-One:

• a) Mobile phone number: One-to-one if each student has a unique phone number.
• b) Student identification number: One-to-one (since IDs are typically unique).
• c) Final grade in the class: One-to-one if no two students have the same grade.
• d) Home town: One-to-one if each student comes from a unique town.

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Would you like more details on any of these conditions or have any other questions? Here are some related questions for further exploration:

1. What is the difference between a one-to-one function and a onto function?
2. How can you determine if a function is injective from a graph?
3. What are the implications of a function being both one-to-one and onto (bijective)?
4. Can a function be one-to-one in one context and not in another? (e.g., mobile numbers in different countries)
5. How do you prove that a given function is not one-to-one?

Tip: When considering functions, check if any two elements of the domain can map to the same element in the codomain to determine injectivity.