elimination

Elimination is the act or process of removing, discarding, or excluding something. The term is used across disciplines to describe methods, outcomes, or rules that reduce a system to fewer components, or to identify what remains after certain elements are removed.

In mathematics and formal logic, elimination refers to procedures for removing variables or quantifiers to simplify problems or derive conclusions. Gaussian elimination solves linear systems by row operations; elimination theory deals with removing variables from polynomial equations; quantifier elimination produces equivalent statements without existential or universal quantifiers.

In science and medicine, elimination denotes the removal of substances from a living organism or environment. Biologically, it refers to excretion of waste products (such as urine and feces). In pharmacology, drug elimination describes the loss of a compound from the body through metabolism and excretion, often summarized by concepts like clearance and elimination half-life. Elimination diets temporarily remove suspected foods to identify triggers of adverse reactions.

In chemistry, elimination is a type of reaction in which atoms or groups are removed from a molecule, forming a new multiple bond, as in dehydrohalogenation or dehydration reactions. In pharmacology and toxicology, elimination also refers to the overall process that reduces the concentration of a substance in the body.

In sports and competitive contexts, elimination means removing a participant or team from a tournament, typically through knockout rounds or after losses, thereby determining advancing or final standings.

Overall, elimination captures the idea of reducing a system by removing elements, whether in mathematics, science, medicine, chemistry, or competition, and it is often accompanied by a corresponding result or simplified structure.