In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces).
[1][2] The noun triviality usually refers to a simple technical aspect of some proof or definition.
[1][3] The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove.
[2] The judgement of whether a situation under consideration is trivial or not depends on who considers it since the situation is obviously true for someone who has sufficient knowledge or experience of it while to someone who has never seen this, it may be even hard to be understood so not trivial at all.
So, triviality is not a universally agreed property in mathematics and logic.
In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure.
These include, among others: "Trivial" can also be used to describe solutions to an equation that have a very simple structure, but for the sake of completeness cannot be omitted.
is important in mathematics and physics, as it could be used to describe a particle in a box in quantum mechanics, or a standing wave on a string.
[4] Similarly, mathematicians often describe Fermat's last theorem as asserting that there are no nontrivial integer solutions to the equation
Trivial may also refer to any easy case of a proof, which for the sake of completeness cannot be ignored.
Similarly, one might want to prove that some property is possessed by all the members of a certain set.
The judgement of whether a situation under consideration is trivial or not depends on who considers it since the situation is obviously true for someone who has sufficient knowledge or experience of it while to someone who has never seen this, it may be even hard to be understood so not trivial at all.
The following examples show the subjectivity and ambiguity of the triviality judgement.
In some texts, a trivial proof refers to a statement involving a material implication P→Q, where the consequent Q, is always true.
[5] A related concept is a vacuous truth, where the antecedent P in a material implication P→Q is false.