eliminations

Eliminations refer to processes in which items are removed from a set or sequence to achieve a result, reduce complexity, or determine an outcome. The term is used across disciplines with related but distinct meanings.

In mathematics and logic, elimination describes methods for removing variables or quantifiers to obtain simpler descriptions. Gaussian elimination reduces systems of linear equations to row-echelon form, enabling solution by back-substitution. In algebraic geometry, elimination theory uses resultants or Grobner bases to eliminate variables and obtain relations among remaining ones. In logic, quantifier elimination transforms formulas by removing existential or universal quantifiers while preserving truth in a given theory.

In statistics, machine learning, and data analysis, variable or feature elimination (often via recursive feature elimination) removes candidate predictors to reduce dimensionality, improve model interpretability, or prevent overfitting.

In competitions and decision-making, elimination denotes the removal of participants or options as a process to reach a winner or a final selection. Single-elimination tournaments proceed by discarding half the participants after each round; the last remaining competitor is the winner. Double-elimination formats allow a participant to lose once before being eliminated. In some voting systems, rounds of elimination remove the lowest-ranked candidates until a winner emerges; this is known as elimination or exhaustive elimination in certain preferential voting methods.

In chemistry, elimination reactions remove atoms or groups from adjacent carbon atoms, forming double or triple bonds. Common types include E1 and E2 mechanisms, which differ in the sequence of bond-breaking and bond-forming steps and in their dependency on substrate structure and conditions.

The concept of elimination thus spans methods for simplifying problems, filtering options, and deriving outcomes across scientific, mathematical, and practical contexts.