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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

### define

Define means to give a definition.

##### Examples

¨  “We define n to be even if it is divisible by  2.”

¨  “Let us define n to be even if it is divisible by  2.”

¨  Define n to be even if it is divisible by  2.”

Some students find the third form  define as a command  to be confusing:  “What am I supposed to do when it tells me to define something?”  Answer:  You don’t do anything, you simply understand that from now on “even” means “divisible by 2”.

### degenerate

The word degenerate is used in some disciplines to describe a particular math structure with one or both of the following properties:

a)     Some parts of the structure that are normally distinct are the same in this particular example.

b)     Some parameter is 0.  (Trivial is also sometimes used for this meaning.)

 More than half of the examples I turned up on JSTOR when doing research for the Handbook were references to degenerate critical points.
##### Examples

¨  A line segment is a degenerate isosceles triangle.  The two equal sides coincide and the third side has length 0.  This fits both a) and b).

¨  A critical point is degenerate if it satisfies a certain technical condition, namely that the determinant of its Hessian is 0.  This fits b).  A small perturbation turns a degenerate critical point into several critical points, so this also “sort of” fits a).

##### Usage

The word degenerate is given a mathematical definition in some fields and is used informally in others.  Many specialties in math never use the word.

### disjoint

Two sets S and T are disjoint if their intersection is empty, in other words if they have no elements in common.  A family of sets is pairwise disjoint if any two different sets in the family are disjoint.

##### Examples

¨  {1,2} and {3,4,5} are disjoint.

¨  Let  denote the set of all positive real numbers and  the set of all negative real numbers.  Then  and  are disjoint.  There is no real number that is both positive and negative.

¨  The set  is  a pairwise disjoint family of subsets of the reals (here (n, n+1) denotes the open interval, not the ordered pair.)

¨   A statement such as "Let A, B and C be disjoint sets” usually means that the sets are pairwise disjoint.  When I was searching the literature for the Handbook, I could not find a clear example where such a statement meant that no element was in every set; it always meant no element was in any two different sets.

##### Warning

“Disjoint” requires two sets.  Don’t say things such as: "Each set in a partition is disjoint".   You could  say “each set in a partition is disjoint from each of the other sets.”

### domain

A domain may be any of these:

¨  The domain of a function.

¨  A connected open set in a topological space.

¨  A type of lattice studied in denotational semantics.

¨  A type of ring, more properly called an integral domain.

I recall as a graduate student being puzzled by the first two meanings given above, with the result that I spent a (mercifully short) time trying to prove that the domain of a continuous function had to be a connected open set.