abstractmath.org

GLOSSARY

# A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

### 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.

### 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.

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.