Examples of the Mathematical Red Herring principle

Solution 1:

All differential equations are stochastic differential equations,

but most stochastic differential equations are not differential equations.

Solution 2:

My understanding of this principle is that sometimes, adjectives widen the scope of nouns (or modify their scope in other, more complicated ways) and this can be confusing. Examples:

partial functions aren't necessarily functions
non-unital rings aren't necessarily rings
non-associative algebras aren't necessarily algebras (under my preferred definition)

Another funny one is:

a partially ordered set isn't necessarily an ordered set.

In this case, an adverb (partially) is widening the scope of an adjective (ordered).

There's a related phenomenon whereby we give a black-box meaning to phrases of the form [adjective]-[noun], and that meaning isn't a compound of the meanings of these two words individually. E.g.

Topological spaces aren't "spaces" because the term "space" lacks a technical meaning
Lawvere theories aren't "theories" because the term "theory" lacks a technical meaning
etc.

Solution 3:

A "set of measure zero" is often defined without saying what measure is used, or what value it takes on the set. Thus, neither the "measure" nor the "zero" are defined/true on their own.