Examples

• In mathematics
  Mathematics
  Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
  
  , it is often important to find factors
  Divisor
  In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
  
   of an integer number N. Any number N has four obvious factors: ±1 and ±N. These are called "trivial factors". Any other factor, if any exist, would be called "nontrivial".

• The matrix
  Matrix (mathematics)
  In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
  
   equation AX=0, where A is a fixed matrix, X is an unknown vector, and 0 is the zero vector, has an obvious solution X=0. This is called the "trivial solution". If it has other solutions X≠0, they would be called "nontrivial"

• In the mathematics of group theory
  Group theory
  In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
  
  , there is a very simple group with just one element in it; this is often called the "trivial group". All other groups, which are more complicated, are called "nontrivial".

• Database theory
  Database theory
  Database theory encapsulates a broad range of topics related to the study and research of the theoretical realm of databases and database management systems....
  
   has a concept called functional dependency
  Functional dependency
  A functional dependency is a constraint between two sets of attributes in a relation from a database.Given a relation R, a set of attributes X in R is said to functionally determine another attribute Y, also in R, if, and only if, each X value is associated with precisely one Y value...
  
  , written . It is obvious that the dependence is true if Y is a subset
  Subset
  In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
  
   of X, so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial".

• The differential equation with boundary conditions is important in math and physics, for example describing a particle in a box
  Particle in a box
  In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
  
   in quantum mechanics, or standing wave
  Standing wave
  In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
  
  s on a string. It always has the solution . This solution is considered obvious and is called the "trivial" solution. In some cases, there may be other solutions (sinusoids), which are called "nontrivial".