Which ordered pairs need to be added to the empty relationp = {}on the set X = {a,b,c,d} to create the symmetric closure p^s of p?

To create the symmetric closure $p^s$ of the empty relation $p = \emptyset$ on the set $X = \{a, b, c, d\}$, we need to follow these steps:

1. Identify the empty relation: The given relation $p$ is empty, meaning it contains no ordered pairs.

2. Understand symmetric closure: The symmetric closure of a relation $p$ is the smallest symmetric relation that contains $p$. A relation is symmetric if for every $(x, y) \in p$, the pair $(y, x)$ is also in $p$.

3. Add pairs to make it symmetric: Since $p$ is empty, there are no pairs to consider. Therefore, the symmetric closure of an empty relation is also an empty relation.

4. Result: The symmetric closure $p^s$ of the empty relation $p$ on the set $X$ is still an empty relation. No ordered pairs need to be added.

Thus, the ordered pairs that need to be added to the empty relation $p = \emptyset$ to create the symmetric closure $p^s$ are none. The symmetric closure remains $\emptyset$.