Under is used to name the function or relation just referred to in the sentence. The reference may be indirect or implicit.
¨ "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.
¨ 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.”
¨ Unit element or unity most likely means an identity element. This requires lexicographical research.
One or more 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.
¨ 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.
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.
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.
¨ An indefinite integral is determined up to a constant. In this case the equivalence relation is that of differing by a constant.