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partition

The word partition has several distinct meanings in math.

A partition of a set into disjoint blocks (same as an equivalence relation).

A partition of an integer, expressing the integer as a sum of integers.

A partition of an interval, used in the theory of integration.

A partition of unity of a topological space.

All of these ideas involve the concept of breaking something up into distinct pieces, but they are genuinely distinct ideas.

permutation

Permutation is defined in the literature in two different ways:

A permutation of an n-element set is a sequence of length n in which each element of the set appears once. This is discussed in detail in Mathworld.

A permutation of a set is a bijection from the set to itself.

Of course, the two definitions can be converted into each other, but psychologically they are rather different. The discussion in Wikipedia gives both definitions.

positive

In most texts and university courses, the phrase " x is positive" means x > 0. In a few texts and in France (and possibly other places in Europe), it may mean .

Acknowledgment

The members of the RUME discussion list.

power

The integer 125 is a power of 5 with exponent 3. You may also describe 125 as " 5 to the third power", or ask, “What power of 5 is 125?” So the word power can refer either to the number 125 or to the exponent 3. Isn’t that silly?

prime

A prime is a typographical symbol, and also a positive integer greater than 1 that has exactly two positive integer divisors (Wi, MW).

proper

If a subset T of a set S is not S, then it is a proper subset of S. (See also here.) This is also used with substructures of a structure (proper subgroup, and so on).

Some authors require that a proper subset of S be not only not all of S but also not be empty. Unless the author makes it clear, you cannot tell which is meant.

The word proper is also used by some authors to mean nontrivial; for example, a proper automorphism would be a non-identity automorphism.

proposition

This word is used in conflicting ways in mathematical writing and in mathematical logic.

In mathematical writing, proposition is another name for theorem, a statement that has been proved true.

In mathematical logic, proposition is comon terminology for a sentence in math English or the symbolic language that is either true or false (here we use the word statement for this).

Posted 31 December 2008