📂Lemmas

The Expression of Trivial in Mathematics

Terminology

In mathematics, when prior definitions or explanations already make the logical context sufficiently clear so that no further detailed explanation is necessary, we call it “trivial”.

Explanation

The expression ’trivial’ is not an everyday word, so many encounter it for the first time while studying mathematics. Using the hanja characters 自 (“self”) and 明 (“bright”), literally translated it means “clear by itself”. The Standard Korean Dictionary explains it as follows.

adjective

1. So obvious that it can be known without explanation or proof.

In mathematics one does not call something trivial while omitting explanations and proofs without basis. As described above, we call something trivial when the explanations and logical implications are already sufficient, or when it is so easy or simple that no explanation is needed.

As suggested by the words “easy” and “sufficient”, there is no absolute standard for triviality. Because it is a convenient expression to use when one is reluctant to repeat an already given argument, readers may find that a statement labeled “trivial” is not trivial for them and requires some thought. Consequently, texts that use this expression often tend to be advanced.

When something is not trivial we call it nontrivial, and the noun form is triviality.

The uniqueness is a triviality.¹

The only nontrivial point is completeness.²

The proof is nontrivial; see Exercise 24.³

Difference from “obvious”

Nothing in the world is truly “obvious”, and this is especially true in mathematics. Saying something is “obvious” carries the nuance of “inevitable” or “bound to be so”, whereas calling something “trivial” typically means “it follows directly from the definition”, “it is very easy”, or “the explanation is already sufficient”.

Examples

By the explanation above, we use “trivial” when one can see at a glance that something is correct to the extent that no explanation is required. When an object carries the label trivial it often refers to simple, easy cases such as the number zero, a constant, the zero vector, the empty set, or a singleton set.

trivial solution

A “trivial solution” is a solution that can be seen immediately without complicated calculations or long arguments. For example, for a polynomial with no constant term, it is trivial that $x = 0$ is one of the roots.

$$ x^{3} -2x^{2} + 4x = 0 $$

Or when finding $\mathbf{x}$ that satisfies the system of equations $\mathbf{x} = \mathbf{0}$, $\mathbf{x} = \mathbf{0}$ is called trivial.

$$ A \mathbf{x} = \mathbf{0} $$

Similarly, for a differential equation like $y(x) = c$, the constant function $y(x) = c$ is immediately seen to be a solution without further explanation, and is called a trivial solution.

$$ ay^{\prime \prime} + by^{\prime} = 0 $$

trivial case

When proving a theorem by breaking it into cases, we call a case trivial if it is immediately evident that the theorem holds in that case.

We prefer to ignore this trivial case.⁴

trivial group

The ordered pair $(\left\{ e \right\}, \cdot)$ formed by the binary operation $\cdot$ defined on the sets $\left\{ e \right\}$ and $e \cdot e = e$ is the most obvious example of a group. Hence this is called the trivial group.

trivial normal subgroup

A subgroup $H$ of a group $G$ is called a normal subgroup if it satisfies

$$ gH = Hg \quad \forall g \in G $$

But by the definition of the identity element it is easy to see that $H = \left\{ e \right\}$ is a normal subgroup. Therefore $\left\{ e \right\}$ is called the trivial normal subgroup.