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Posted 16 April 2009

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factor

fallacy

A fallacy is an error in reasoning. Two fallacies with standard names that are commonly committed by students are affirming the consequent and denying the hypothesis. See also argument by analogy and Example 1 under conditional assertion

Terminology The meaning of fallacy given here is that used in this Handbook. It is widely used with a looser meaning and often connotes deliberate deception, which is not intended here.

family

A family of sets sometimes means an indexed set (tuple) of sets (so differently indexed members may be the same) and sometimes merely a set of sets.

field

¨  A field is a function assigning values (usually denoting a physical quantity) to points in a space.

¨  A field is an algebraic structure allowing addition, subtraction, multiplication, and non-zero division subject to certain laws.

 The two meanings are unrelated, and you usually have to tell from the context which one is meant. 

The word also refers to specialty, as in “My field is topological semigroups”.  This meaning occurs in all academic subjects.

find

Used in problems in much the same way as give.

fix

¨  A function f fixes a point p if f(p) = p. This is based on this metaphor: you fix an object if you make it hold still (she fixed a poster to the wall). In my observation, Americans rarely use “fix” this way; in the USA, the word nearly always means “repair”.

¨  “Fix” is also used in sentences such as “In the following we fix a point p one unit from the origin”, which means that we will be talking about any point one unit from the origin (a variable point!) and we have established the notation p to refer to that point. The metaphor behind this usage is that, because it is called p, every reference to p is to the same value (the value is “fixed” throughout the discussion.)

follow

¨  The statement that an assertion Q follows from an assertion P means that P implies Q (if P, then Q).

Example

“The integer n  is divisible by 16.  It follows that it is even.”

¨  The word “follow” is also used to indicate that some statements after the current one are to be grouped with the current one, or (as in “the following are equivalent)” are to be grouped with each other.

Example

 “A set G with a binary operation is a group if it satisfies the following axioms ... ” This statement indicates that the axioms that follow are part of the definition currently in progress.

following are equivalent

The phrase “the following are equivalent” (or “TFAE”) is used to assert the equivalence of the following assertions (usually more than two and presented in a list).

for all

See universal quantifier.

formula

¨  In informal usage in math, a formula is an expression allowing you to calculate some quantity.   More here.

¨  In math logic, a formula is an arrangement of symbols that represents a statement, possibly containing variables.   Formulas are almost always defined in terms of a particular formal language (MW, Wi).

The equation  is a formula in both senses, but  (“the formula for the area of a circle”) is a formula only in the first sense above.  In mathematical logic,  would be called a term.

functional