The task is asking to identify the type of functions represented by the arrow diagrams from four different mappings (a, b, c, d). Specifically, we need to determine whether the functions are Surjective (Onto), Injective (One-to-One), or Bijective (both Injective and Surjective).

Let's review each option:

1. Option a:

   • The function maps element 1 to both "a" and "b," element 2 to "b," and element 3 to "c."
   • This function is neither injective nor surjective.
     • Not injective: Element 1 is mapped to both "a" and "b," violating the one-to-one property.
     • Not surjective: "a" is not mapped by any element.

2. Option b:

   • The function maps element 1 to "a," element 2 to "b," and element 3 to "c."
   • This function is surjective but not injective.
     • Surjective: Every element in the codomain (a, b, c) has a preimage.
     • Not injective: Two elements from the domain, 1 and 2, are mapped to the same element ("a"), violating injectivity.

3. Option c:

   • The function maps element 1 to "a," element 2 to "a," and element 3 to "b."
   • This function is surjective but not injective.
     • Surjective: All elements in the codomain (a, b) are covered.
     • Not injective: Elements 1 and 2 map to the same element ("a").

4. Option d:

   • The function maps element 1 to "a," element 2 to "b," element 3 to "c," and element 4 to "d."
   • This function is bijective.
     • Injective: Every element in the domain maps to a unique element in the codomain.
     • Surjective: All elements in the codomain are mapped by elements from the domain.

Final Answer:

• Option a: Neither injective nor surjective.
• Option b: Surjective but not injective.
• Option c: Surjective but not injective.
• Option d: Bijective (injective and surjective).

Would you like any further details or clarifications?

Related Questions:

1. What is the formal definition of an injective function?
2. How can you tell if a function is surjective from its mapping diagram?
3. What are the conditions for a function to be bijective?
4. Why is it impossible for a function to be both injective and surjective if the domain has fewer elements than the codomain?
5. How can we visually determine injectivity and surjectivity in a function diagram?

Tip: Remember that bijective functions must pass both the horizontal and vertical line tests in their diagrams!