Let's analyze the given diagrams in the image based on whether the functions are onto, injective, or bijective.

1. Onto Function (Surjective): Every element in the codomain (right set) must have at least one pre-image in the domain (left set). All elements on the right should be mapped to.

2. Injective Function (One-to-One): Every element in the domain (left set) maps to a unique element in the codomain (right set). There shouldn't be multiple elements from the domain pointing to the same element in the codomain.

3. Bijective Function: The function is both injective and onto, meaning there is a perfect one-to-one correspondence between the domain and codomain. Each element from the domain maps uniquely to an element in the codomain, and all elements in the codomain are covered.

Now, analyzing each diagram:

• Diagram a:

  • This is not injective because two elements from the domain map to the same element in the codomain.
  • It is onto since every element in the codomain is mapped to by at least one element from the domain.
  • Therefore, it is surjective but not bijective.

• Diagram b:

  • This is injective since each element from the domain has a unique image in the codomain.
  • It is not onto, as one element in the codomain is not mapped by any element in the domain.
  • Hence, it is injective but not surjective (not bijective).

• Diagram c:

  • This is injective as each element in the domain has a unique image.
  • It is also onto because every element in the codomain has a pre-image.
  • Therefore, it is bijective (both injective and surjective).

• Diagram d:

  • This is not injective because two elements in the domain map to the same element in the codomain.
  • It is onto since every element in the codomain is mapped to by at least one element from the domain.
  • So, it is surjective but not bijective.

• Diagram e:

  • This is injective since each element in the domain maps to a unique element in the codomain.
  • It is not onto, as not all elements in the codomain have pre-images.
  • Therefore, it is injective but not surjective (not bijective).

• Diagram f:

  • This is injective since all elements in the domain have unique images.
  • It is also onto, as every element in the codomain is mapped to.
  • Hence, it is bijective.

Summary:

• a: Surjective (onto).
• b: Injective.
• c: Bijective.
• d: Surjective (onto).
• e: Injective.
• f: Bijective.

Would you like further clarification on any of these classifications?

Here are some related questions to extend your understanding:

1. What are the differences between surjective and injective functions?
2. How do we prove that a function is bijective using algebraic methods?
3. Can a function be neither surjective nor injective? Provide examples.
4. Why is the identity function considered bijective?
5. How do we apply the concept of bijection in real-world problems?

Tip: A bijective function always has an inverse, which can be very useful when working with compositions of functions!