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GLOSSARY
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Define means to give a definition.
¨ “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
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.)
¨ 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).
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.
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.
¨ {1,2} and {
¨ Let
¨ The set
¨ 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.
“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.”
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.