trivially

Trivially is an adverb derived from the adjective trivial, used to indicate that something is done with little effort, or that a statement is self-evident or follows immediately from definitions or basic reasoning. In ordinary language, to do something trivially means to do it easily or without special skill. In formal contexts, especially mathematics, a claim described as trivially true or trivially false rests on the structure of the statement rather than on a deep argument. For example, 2 + 2 = 4 is trivially true under the standard interpretation of arithmetic. The identity map on any algebraic structure is trivially a homomorphism, and the empty function is trivially a function.

Etymology: trivially comes from Latin trivialis 'commonplace, ordinary', from triv- 'three roads' via the concept of a crossing; it entered English through Old French or directly from Latin.

In mathematics and logic, the term is used to distinguish nontrivial results—those requiring substantial argument—from those that follow immediately by definition or a one-line deduction. The adjective 'trivial' is used similarly to denote objects with no important structure, such as the trivial group (containing a single element). The phrase 'trivially satisfied' refers to a condition that holds for all cases without further check.