This is an update of an article in the abstractmath glossary.

About theorems

A theorem is said to be trivial to prove or trivially true

• a) If the theorem follows by rewriting using definitions, or
• b) If the speaker’s mental representation of the mathematical object mentioned in the theorem makes the truth of the fact immediately perceivable.   

Example:

Here is a scenario that exemplifies (a):

• A textbook defines the image of a function $F:A\to B$ to be the set of all elements of $B$ of the form $F(a)$ for some $a\in A$.
• It then goes on to say that $F$ is surjective if for every element $b$ of $B$ there is an element $a\in A$ with the property that $F(a) = b$.
• It then states a theorem, or give an exercise, that says that a function $F:A\to B$ is surjective if and only if the image of $F$ is $B$.
• The proof follows immediately by rewriting using definitions.
• The instructor calls the proof trivial and goes on to the next topic.
• Some students are totally baffled.

I have seen this happen many times with this and other theorems.  This sort of incident may be why many intelligent people feel they are "bad at math".

People are not born knowing the principle of rewriting by definitions. The principle needs to be TAUGHT.

• When a class is first introduced to proof techniques the instructor should explicitly describe rewriting by definitions with several examples.
• After that, the instructor can say that a proof follows by rewriting by definitions and make it clear that the students will have to do the work (then or later).
• Such a proof is justly called "trival" but saying it is trivial is also a putdown if no one has pointed out the procedure of rewriting by definitions.

Example:

This example illustrates (b).

Theorem: Let $G$ be a finite group and $H$ a subgroup of index $2$ (meaning it has half the number of elements of the group).  Then $H$ is normal in $G$.

Basic facts about groups and subgroups learned in first semester abstract algebra:

• A subgroup of a group determines a partition consisting of left cosets and another partition of right cosets, each (in the finite case) with the same number of elements as the subgroup.
• A subgroup is a left coset of itself and also a right coset of itself.
• If every left coset is also a right coset and vice versa (so the two partitions just mentioned are the same), then by definition the subgroup is normal in the group.

Now if $H$ has index $2$ that means that each partition consists of two cosets. In both cases, one of them has to be $H$, so the other one has to be $G\setminus H$, which must therefore be a left and right coset of $H$. So $H$ is normal in $G$.

So once you understand the basics about cosets and normal subgroups, the fact that $H$ has to be normal if it is of index $2$ is "obvious". I don't think you should call this "trivial". Best to say it is "obvious if you have a clear understanding of cosets of groups".

About mathematical objects

• A function may be called trivial if it is the identity function or a constant function, and possibly in other circumstances. (If someone showed that the cosmo­logical constant is 0 that would not be called trivial.)
• A solution to an equation may be said to be trivial if it is 0 or 1. There may be other situations in which a solution is called "trivial" as well.
• A mathematical structure is said to be trivial if its underlying set is empty or a singleton set. In particular, a subset of a set is nontrivial if it is nonempty. I have not found an example where "nontrivial subset" means it is not a singleton. 

Note: "Trivial" and "degenerate" overlap in meaning but are not interchangeable.  What is called "degenerate" seems to depend on the mathematical specialty.