# Nontrivial

Nontrivial is the opposite of trivial. In contexts where trivial has a formal meaning, nontrivial is its antonym.

It is a term common among communities of engineers and mathematicians, to indicate a statement or theorem that is not obvious or easy to prove.

## Examples

• In mathematics, it is often important to find factors 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".[1]
• The matrix 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"[2]
• In the mathematics of group theory, 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".
• In the graph theory the trivial graph is a graph which has only 1 vertex and no edges.
• Database theory has a concept called functional dependency, written $X \to Y$. It is obvious that the dependence $X \to Y$ is true if Y is a subset of X, so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial".
• The differential equation $f''(x)=-\lambda f(x)$ with boundary conditions $f(0) = f(L) = 0$ is important in math and physics, for example describing a particle in a box in quantum mechanics, or standing waves on a string. It always has the solution $f(x) = 0$. This solution is considered obvious and is called the "trivial" solution. In some cases, there may be other solutions (sinusoids), which are called "nontrivial".[3]
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