We are given the sets $A = \{1\}$ and $B = \{1, 2, 3\}$, along with the relation $R \subseteq A \times B$ defined as:

$R = \{(1, 1), (1, 2), (1, 3)\}$

Let's evaluate the given options.

1. Is $R$ injective?

   A relation is injective (one-to-one) if no two distinct elements in $A$ are mapped to the same element in $B$. Here, we only have one element in $A$ (which is $1$), and it is related to three different elements in $B$ (namely $1, 2,$ and $3$). Since the same element in $A$ is mapped to multiple elements in $B$, the relation is not injective.

2. Is $R$ surjective?

   A relation is surjective (onto) if every element of $B$ has a preimage in $A$. In this case, all elements of $B$ (1, 2, and 3) are mapped from the element $1 \in A$. Therefore, the relation is surjective because every element in $B$ is covered.

3. Is $R$ a function?

   A relation is a function if every element of $A$ is related to exactly one element in $B$. However, in this case, the element $1 \in A$ is related to multiple elements in $B$ (1, 2, and 3). This violates the definition of a function, so $R$ is not a function.

4. None of the other options?

   Since the relation is surjective but neither injective nor a function, this option does not apply.

Conclusion:

• $R$ is surjective.
• $R$ is not injective.
• $R$ is not a function.

Would you like further clarification on any of these concepts?

Related questions:

1. Can a relation be injective but not surjective?
2. What conditions must a relation satisfy to be a function?
3. How can we visualize injective and surjective relations using diagrams?
4. Can a function be neither injective nor surjective?
5. How would the answer change if $A$ had more elements?

Tip:

For a relation to be a function, each input (from set $A$) must correspond to exactly one output (from set $B$).