Are the following sets of ordered pairs functions ? If so, examine whether the mapping is surjective or injective :
{(a, b) : a is a person, b is an ancestor of a}

Answer

To determine whether the set of ordered pairs \(\{(a, b) : a \text{ is a person, } b \text{ is an ancestor of } a\}\) represents a function, we need to analyze the definition of a function and the nature of the relationship between the elements \(a\) and \(b\).

### Step-by-Step Solution:

1. **Definition of a Function**:
A set of ordered pairs is considered a function if each element in the domain (the first component of the pairs) is associated with exactly one element in the codomain (the second component of the pairs).

2. **Identify the Domain and Codomain**:
In this case, the domain consists of all persons \(a\), and the codomain consists of all possible ancestors \(b\) of those persons.

3. **Examine the Relationship**:
For a given person \(a\), there can be multiple ancestors \(b\). For example, a person can have a mother, father, grandparents, etc. This means that for a single input \(a\), there are multiple possible outputs \(b\).

4. **Conclusion on Functionality**:
Since a single person \(a\) can be associated with multiple ancestors \(b\), this violates the definition of a function, which requires that each input has a unique output. Therefore, the set of ordered pairs does not represent a function.

5. **Surjectivity and Injectivity**:
Since the set is not a function, we cannot examine whether it is surjective (onto) or injective (one-to-one). These properties only apply to sets that qualify as functions.

### Final Answer:
The set \(\{(a, b) : a \text{ is a person, } b \text{ is an ancestor of } a\}\) is **not a function** because a single person can have multiple ancestors, leading to multiple outputs for a single input.