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under

Under is used to name the function or relation just referred to in the sentence. The reference may be indirect or implicit.

Examples

¨ "The set of integers is a group under addition."

¨ "If x is related to y under the relation E, we write x E y."

¨ "Let F and G be functions defined on the real numbers. If the value of x under F is greater than the value of x under G for every x, then we say that F > G." This can be translated as: “If for every x, then we say that F > G.”

It is my impression that some mathematicians use “under” in this way a lot, but most mathematicians don’t use it at all. This needs more lexicographical research than what I did for the Handbook.

underlying set

unique

¨ To say that an object satisfying certain conditions is unique means that there is only one object satisfying those conditions. For example, there is a unique even prime, namely the integer 2.

¨ The Handbook, page 259, discusses the philosophical confusion connected with questions such as “Is there a unique set of integers”. Mathematicians normally talk as if there is a unique set, but when pressed by foundations questions may say things like “Well, there are many copies but let’s assume we have picked a particular one.”

¨ The word "unique" is misused by students; this is discussed here. See also up to.

unit, unit element, unity

¨ A unit in a ring may mean either the identity element of the ring, or an invertible element of the ring. However, “ring with unit” means ring with identity element.

¨ Unit element or unity most likely means an identity element. This requires lexicographical research.

unknown

One or more variables may occur in a constraint, and the intent of the discourse may be to determine the values of the variables that satisfy the constraint. In that case the variables may be referred to as unknowns.

Examples

¨ Find the values of x for . Answer: .

¨ Find the values of x for which . Answer: .

In both these problems x would be called an unknown.

unpack, unwind

A typical definition in mathematics may make use of a number of previously defined concepts. To unpack or unwind such a definition is to replace the defined terms with explicit, spelled-out requirements. See translation problem and rewrite using definitions.

Similarly a function may be defined by a complicated formula. To unpack such a formula means investigating it piece by piece, or chunk by chunk. Zooming and Chunking has an example, and Equivalence Relations has another one.

up to

Let E be an equivalence relation. To say that a definition or description of a type of mathematical object determines the object up to E (or modulo E) means that any two objects satisfying the description are equivalent with respect to E.

Examples

¨ An indefinite integral is determined up to a constant. In this case the equivalence relation is that of differing by a constant.

¨ The statement "G is a finite group of order n containing an element of order n" forces G to be the cyclic group of order n, so that the statement defines G up to isomorphism.