1. The expression $2 |A \cap B|$ involves the cardinality (size) of the intersection of two sets $A$ and $B$. 2. The intersection $A \cap B$ represents all elements that are common to both sets $A$ and $B$. 3. The vertical bars $| \cdot |$ denote the number of elements in a set, so $|A \cap B|$ is the count of elements in both $A$ and $B$. 4. The factor 2 in front of $|A \cap B|$ often appears in formulas involving set operations, such as the principle of inclusion-exclusion, where it accounts for counting the intersection twice when summing $|A| + |B|$. 5. For example, the formula for the union of two sets is: $$|A \cup B| = |A| + |B| - |A \cap B|$$ 6. If you see $2 |A \cap B|$, it might be part of a rearranged or extended formula where the intersection is counted twice for correction or other purposes. 7. To understand exactly where $2 |A \cap B|$ comes from, consider the context or formula you are working with, such as: $$|A| + |B| = |A \cup B| + |A \cap B|$$ which can be rearranged to include $2 |A \cap B|$ in some steps. In summary, $2 |A \cap B|$ comes from counting the elements in the intersection of $A$ and $B$ twice, often in the context of set cardinality formulas like inclusion-exclusion.