Suppose 𝑓:𝐴→𝐴
 is a function. Mark Y next to anything that means the same as saying 𝑓
 is a permutation and N next to anything that does not.

  1. One-to-one

  2. In the schematic, at least one arrow out of each point

  3. In the schematic for functions, at least one arrow into each point

  4. In the schematic, exactly one arrow out of each point

  5. Onto

  6. Injective

  7. Surjective

  8. Injective and surjective

  9. In the schematic for functions, at most one arrow into each point

  10. The inverse image of each element in the codomain has cardinality at least one

To determine which of the statements align with the concept of a **permutation**, we should recall that a permutation is a bijection (both injective and surjective) from a set \(A\) to itself. That is, each element in the set must map to exactly one unique element, and every element must have a preimage.

Let's mark Y for statements that correspond to this and N for those that don't:

1. **One-to-one**: Y  
   - This is another way to describe injectivity, which is required for a permutation.

2. **In the schematic, at least one arrow out of each point**: N  
   - This describes surjectivity, but surjectivity alone is not enough for a permutation.

3. **In the schematic for functions, at least one arrow into each point**: N  
   - This indicates surjectivity, but doesn't cover injectivity, which is also necessary for a permutation.

4. **In the schematic, exactly one arrow out of each point**: Y  
   - This indicates a well-defined function where each element of the domain is mapped exactly once, crucial for a permutation.

5. **Onto**: Y  
   - "Onto" refers to surjectivity, which is required for a permutation.

6. **Injective**: Y  
   - A permutation must be injective, meaning no two elements map to the same image.

7. **Surjective**: Y  
   - A permutation must be surjective, meaning every element in the codomain has a preimage.

8. **Injective and surjective**: Y  
   - A permutation is both injective (one-to-one) and surjective (onto), so this is correct.

9. **In the schematic for functions, at most one arrow into each point**: Y  
   - This describes injectivity, where no element in the codomain is the image of more than one element from the domain, which is required for a permutation.

10. **The inverse image of each element in the codomain has cardinality at least one**: N  
    - This describes surjectivity, but not injectivity, which is also required for a permutation.

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Would you like further details on any of these answers? Here are some related questions you could explore:

1. What are the formal definitions of injective and surjective functions?
2. How do you prove that a function is a permutation?
3. Can a function be injective but not surjective, or vice versa?
4. What are the consequences of a function being bijective?
5. How can permutations be represented in group theory?

**Tip**: Always keep in mind that for a function to be a permutation, it must be both injective and surjective, which guarantees a one-to-one correspondence between elements of the domain and codomain.

Suppose 𝑓:𝐴→𝐴 is a function. Mark Y next to anything that means the same as saying 𝑓 is a permutation and N next to anything that does not. 

1. One-to-one 
2. In the schematic, at least one arrow out of each point 
3. In the schematic for functions, at least one arrow into each point 
4. In the schematic, exactly one arrow out of each point 
5. Onto 
6. Injective 
7. Surjective 
8. Injective and surjective 
9. In the schematic for functions, at most one arrow into each point 
10. The inverse image of each element in the codomain has cardinality at least one.

Explore how to identify whether a function is a permutation by analyzing injective and surjective properties. This problem covers key concepts like bijections, injective functions, surjective functions, and their role in defining permutations. Suitable for students in Grades 10-12.

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