Final Answer:
The cardinality of the union of sets a and b, denoted as |a ∪ b|, is given by the formula |a ∪ b| = |a| + |b| - |a ∩ b|.
Step-by-step explanation:
The formula |a ∪ b| = |a| + |b| - |a ∩ b| is derived from the principle of counting the elements in the union of two sets. The cardinality of a set (|a| or |b|) represents the number of elements in that set. The union of sets a and b contains all distinct elements from both sets. However, to avoid overcounting, we subtract the cardinality of the intersection of a and b (|a ∩ b|). This ensures that elements common to both sets are not counted twice.
For a clearer understanding, consider an example. Let |a| = 3, |b| = 4, and |a ∪ b| = 5. Using the formula, we have 5 = 3 + 4 - |a ∩ b|. Solving for |a ∩ b|, we find that |a ∩ b| = 2. This means there are two elements common to both sets a and b. The formula is a fundamental tool in set theory, helping mathematicians analyze and quantify the relationships between different sets.
In conclusion, the formula |a ∪ b| = |a| + |b| - |a ∩ b| provides a systematic approach to determine the cardinality of the union of two sets. It is a valuable tool in combinatorics and probability theory, allowing mathematicians to calculate the number of distinct elements in complex scenarios involving multiple sets.