Introduction to Set Theory and Logical Operations
Set theory is the branch of mathematical logic that studies sets, which are collections of distinct objects called elements. Formalized by Georg Cantor in the late 19th century, set theory forms the foundational language of modern mathematics, computer science, database queries (SQL joins), and probability. Understanding set operations is essential for analyzing relations, studying logic gates, and filtering arrays in programming.
Sets are typically represented using braces containing comma-separated elements (e.g., $A = \{1, 2, 3\}$). When comparing multiple sets, several operations are used to create new sets based on the relations of their elements. The most common operations are the Union (combining all elements), Intersection (finding shared elements), Difference (finding elements in one set but not another), and Symmetric Difference (finding elements in either set but not both).
This calculator simplifies these set comparisons. By entering two comma-separated lists of elements, the solver computes all standard operations, lists the cardinalities, and displays the distribution on a visual Venn diagram.