distincts

Distincts is a term used to describe the property of being different or not identical. In mathematics and logic, two objects are distinct if they are not the same object; equivalently, they are not equal under the chosen notion of equality. The concept underpins identity and is central to how sets are defined: a set consists of distinct elements, with no element appearing more than once.

In counting and data handling, the number of distinct elements in a collection refers to the size of the set of unique values contained there. Distinctness is often achieved by removing duplicates. For example, the collection {1, 1, 2, 3, 2} has three distinct elements: 1, 2, and 3. Distinctness is relative to an equivalence relation; when equality is used as the relation, x and y are distinct if and only if x ≠ y.

Applications of the notion of distincts appear across disciplines. In databases and programming, operations such as counting distinct values or selecting distinct rows help summarize data and avoid duplicates. In mathematics, distinguishing distinct points or objects is essential in geometry and algebra. In logic and type theory, distinct terms may represent different objects or proofs, even if they share some properties. In all uses, the fundamental idea is to differentiate objects to prevent conflating them and to enable precise counting, comparison, and classification.