The question is asking about the type of function (onto, injective, or bijective) between two sets $A = \{p, q, r, s\}$ and $B = \{1, 2, 3, 4\}$.

Let’s recall the definitions:

1. Injective (One-to-one): A function is injective if each element in the domain maps to a distinct element in the codomain (i.e., no two elements from the domain map to the same element in the codomain).
2. Onto (Surjective): A function is onto if every element in the codomain has at least one element from the domain mapped to it.
3. Bijective: A function is bijective if it is both injective and onto.

Now let's analyze each option.

Option a:

$\{(p, 1), (q, 1), (r, 3), (s, 4)\}$

• Injective? No, because both $p$ and $q$ map to $1$, so it's not injective.
• Onto? Yes, because all elements in set $B = \{1, 2, 3, 4\}$ are mapped.
• Bijective? No, because it’s not injective.

Option b:

$\{(p, 1), (q, 2), (r, 4), (s, 3)\}$

• Injective? Yes, all elements in $A$ map to distinct elements in $B$.
• Onto? Yes, every element in $B = \{1, 2, 3, 4\}$ is mapped.
• Bijective? Yes, because it’s both injective and onto.

Option c:

$\{(p, 3), (q, 2), (r, 1), (s, 4)\}$

• Injective? Yes, all elements in $A$ map to distinct elements in $B$.
• Onto? Yes, every element in $B = \{1, 2, 3, 4\}$ is mapped.
• Bijective? Yes, because it’s both injective and onto.

Option d:

$\{(p, 2), (q, 2), (r, 2), (s, 2)\}$

• Injective? No, because all elements in $A$ map to the same element $2$.
• Onto? No, only element $2$ in $B$ is mapped; elements $1, 3, 4$ are not mapped.
• Bijective? No, because it’s neither injective nor onto.

Conclusion:

• Option a is onto.
• Option b is bijective (both injective and onto).
• Option c is also bijective (both injective and onto).
• Option d is neither injective, onto, nor bijective.

Would you like further details or have any questions?

Related Questions:

1. What is the difference between injective and surjective functions?
2. Can you provide an example of a function that is injective but not surjective?
3. How can we verify if a function is bijective using set mappings?
4. What are the practical applications of bijective functions in mathematics?
5. How does the size of the domain and codomain affect the classification of functions?

Tip:

When verifying if a function is bijective, always check if every element in the domain maps to a unique element in the codomain (injective) and if all elements in the codomain are covered (onto).