Question

(A) Use the analogy of the citizens of the U.S. who have Social Security numbers and the set of Social Security numbers to illustrate bijective maps. (B) Give an analogy for the set of maps which are surjective but not injective.

Short answer

(A) In the context of U.S. citizens and their Social Security numbers, a bijective map would mean that each U.S. citizen with a Social Security number (set A) is assigned a unique and distinct Social Security number (set B), and every Social Security number is also assigned to a U.S. citizen, with no unused Social Security numbers. (B) A surjective but not injective map using the same context would mean that every Social Security number (set B) is assigned to at least one U.S. citizen (set A), but some U.S. citizens may share the same Social Security number. While the function is onto (surjective), it's not one-to-one (injective).

Step-by-step solution

Define bijective, surjective, and injective functions

In order to create analogies for the given sets, we first need to understand what bijective, surjective, and injective functions are. A function from set A to set B is called: 1. Injective (one-to-one) if each element of set A is mapped to a unique (distinct) element in set B, and never to the same element twice. 2. Surjective (onto) if for every element in set B, there's at least one element in set A that maps to it. 3. Bijective if it is both injective and surjective. In a bijective function, there's a perfect one-to-one correspondence between the elements of set A and set B.

Create a bijective map analogy

In the context of U.S. citizens and their Social Security numbers, we can create the following analogy for a bijective map: Set A contains all U.S. citizens who have Social Security numbers. Set B contains all Social Security numbers. A bijective map in this situation would mean that each U.S. citizen with a Social Security number (set A) is assigned a unique and distinct Social Security number (set B). Every Social Security number is also assigned to a U.S. citizen, meaning there are no unused Social Security numbers. This one-to-one correspondence represents a bijective map.

Explain the analogy for the set of maps which are surjective but not injective

An analogy for a map that is surjective but not injective can also be created using the context of U.S. citizens and Social Security numbers. In this scenario, we would have: Set A contains all U.S. citizens who have Social Security numbers. Set B contains all Social Security numbers. A surjective but not injective map in this case would mean that every Social Security number (set B) is assigned to at least one U.S. citizen (set A), but some U.S. citizens may share the same Social Security number. In other words, a U.S. citizen might have the same Social Security number as another U.S. citizen, but every Social Security number is assigned to someone. This would mean that while the function is onto (surjective), it's not one-to-one (injective).

Key concepts

Injective Functions

Injective functions, also known as one-to-one functions, are a fundamental concept in mathematics related to function mappings between sets. For a function to be considered injective from set A to set B, each element in set A must map uniquely to a distinct element in set B. This means no two elements in set A should map to the same element in set B.

Imagine a scenario where each student in a classroom is assigned a unique locker. Here, the set of students represents set A, and the set of lockers represents set B. An injective function would ensure that each student has their own locker, and no two students share the same locker.

You can think of injective functions like setting up a unique email address for each person. Each email address corresponds to only one person, ensuring a clear one-to-one relationship.

• In mathematics, an injective function is formally defined such that if \( f(a_1) = f(a_2) \), then \( a_1 = a_2 \).
• This characteristic ensures that an injective function never assigns the same value in set B to different elements in set A.

Surjective Functions

Surjective functions, also known as onto functions, describe a special type of function mapping where every element in the target set B must be mapped by at least one element from the source set A. In mathematical terms, a function \( f: A \rightarrow B \) is surjective if, for every element \( b \) in set B, there is at least one corresponding element \( a \) in set A such that \( f(a) = b \).

This might be comparable to assigning tasks to a team where each task needs to be completed by at least one member. Set A is the team members, and set B is the tasks. Every task (element of B) must be worked on by someone (element from A), warranting that nothing is left undone.

Another way to visualize it is considering a function that assigns postal codes to addresses within a region. A surjective assignment ensures that each postal code is used for at least one address, covering all possibilities in the region.

• The main requirement is that all elements of set B are accounted for, even if more than one element in set A maps to the same \( b \).
• Hence, a surjective function ensures full coverage of the target set.

Function Mappings

The idea of function mappings is central to understanding how functions provide relationships between elements of two sets. Function mappings can be injective, surjective, or bijective, depending on how they associate elements from set A to set B.

Functions serve as a bridge, linking distinct elements in one set with elements in another. In simpler terms, imagine set A is a group of inputs and set B is the corresponding outputs. Function mapping determines how each input transforms into an output.

• An injective mapping ensures no two inputs result in the same output.
• A surjective mapping guarantees all possible outputs are achieved.
• A bijective mapping combines both, ensuring each input has a unique output and covers all outputs.

Understanding these mappings is crucial for exploring deeper mathematical concepts such as function inverses and compositions.

Just like a vending machine that needs a specific coin (input) to dispense a particular snack (output), function mappings illustrate the diverse relationships and rules between two sets.