# Set Operations Calculator (Union/Intersection/Difference) Union, intersection, difference, symmetric difference, and Cartesian product of two comma-separated sets, with cardinalities and capped pair display. > Canonical page: https://elysiatools.com/en/tools/set-operations-calculator - **Category:** Math & Numbers - **Keywords:** set operations, union, intersection, difference, symmetric difference, cartesian product, set theory, discrete math, cardinality, ordered pairs ## Overview Perform core discrete mathematics set operations on two comma-separated lists, including Union (A ∪ B), Intersection (A ∩ B), Difference (A − B), Symmetric Difference (A △ B), and Cartesian Product (A × B), with automatic deduplication and calculated set cardinalities. ## Inputs - **Set A (comma-separated)** (text): Elements of A as comma-separated text; comparison is case-sensitive and duplicates collapse. - **Set B (comma-separated)** (text): Elements of B as comma-separated text. - **Operation** (select) ## When to use - When solving discrete mathematics or set theory exercises that require calculating set relations and cardinalities. - When comparing two discrete lists of keys, tags, or IDs to determine overlapping or exclusive entries. - When generating all valid ordered pair combinations between two distinct sets of attributes. ## How it works - Enter the elements for Set A and Set B as comma-separated values, which automatically deduplicate while preserving entry order. - Select the required operation from the dropdown: Union, Intersection, Difference, Symmetric difference, or Cartesian product. - Review the resulting output set along with individual and result cardinalities |A|, |B|, and |Result|. ## Use cases - Comparing feature access lists across two user tiers to isolate unique and shared entitlements. - Verifying homework solutions in set theory and discrete math courses. - Generating combinatorial ordered pairs between product sizes and color variants. ## Frequently asked questions ### Are element comparisons case-sensitive? Yes, element comparisons are case-sensitive, meaning 'abc' and 'ABC' are treated as separate elements. ### How does the tool handle repeated values within an input? Duplicate values within the same input set are automatically collapsed, keeping only the first occurrence. ### What is the difference between Difference and Symmetric Difference? Difference (A − B) extracts elements exclusive to Set A, while Symmetric Difference (A △ B) keeps elements that belong to either Set A or Set B, but not both. ### How does the Cartesian product operation work? It generates all ordered pairs (a, b) where each element from Set A is combined with every element from Set B. ### What does cardinality represent in the output? 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