Union and Intersection with the Empty Set

Problem

Consider a set $S$ and the empty set $\emptyset$. Prove the following statements:

1. $S \cup \emptyset = S$

2. $S \cap \emptyset = \emptyset$

Solution

1. $S \cup \emptyset = S$

$S \cup \emptyset = \{x:x \in S \or x \in \emptyset \}$

Since there exists no elements in the empty set, all elements in both sets are just found in the set $S$. Therefore, $S \cup \emptyset = S$.

2. $S \cap \emptyset = \emptyset$

$S \cup \emptyset = \{x:x \in S \and x \in \emptyset \}$

Since there exists no elements in the empty set, there are no elements common to both sets. The absence of elements in a set is by definition the empty set. Therefore, $S \cap \emptyset = \emptyset$.