A glossary of terms

This web page lists various words or terms that might be used in this course. (This list probably contains rather more than you need. Look up a word here when you need it. Don't try and memorise this list.)

aka: Abbreviation for also known as.
axiom: A statement or assertion that will not be proved, but which is to be used as one of the initial assumptions that all proofs are based on.
antecedent: For a statement that is an implication, this denotes the part of that statement to the left of the word or symbol for "implies".
a priori: A useful Latin phrase referring to deductions and meaning from the point of view of correct deductive logic.
bijection: A function $f : A \to B$ which is both an injection and a surjection. Bijections are also called one-to-one correspondences.
bound variable: A variable in an expression or statement that is thought to range over all values from a particular set, such as over all integers, or over all real numbers, etc. The underlying set in question will either be specified in the notation or more usually will be implicitly understood. For example, $r$ in $\sum_{r = 1}^{\infty} \frac{1}{2^{r}}$ or $n$ in the sequence $(a_{n})_{n = 1, 2, \dots}$ . (Note: the $n$ in the common abbreviation the sequence $(a_{n})$ is a dummy variable ranging over all positive integers. The $n$ in the value ${a_{n}}^{2}$ usually has a specific value and is not a dummy; here, the $n$ th term in the sequence has been taken and squared.)
canonical: Meaning according to a set of rules, or canon. This word is used in mathematics to mean that the definition of the object in question is forced upon us in some natural way: either because it is the simplest or most natural such definition that works or (more usually) because it is the only such definition. (The word canon has nothering to do with the word cannon, which means a kind of gun.)
compare and constrast: A classic phrase in an essay title meaning to find as many points of similarity, and points of difference, between two things, and to draw whatever conclusions you feel are appropriate (with reasons).
discrete: Data in mathematics is discrete if it can only take certain values that are clearly separated from other values. For example the integers form a discrete system of numbers because $n$ and $n + 1$ are always clearly separated (there is no number between) but the reals do not form a discrete system of numbers since for example there is no gap that separates $0$ from all of $0.1, 0.01, 0.001, 0.0001, 0.00001, \dots$ . The word "discrete" is not the same as "discreet", which means quiet or unobtrusive.
discuss/discussion: A word often used to indicate or initiate an essay or exploration without prejudging the answer either way. For example, in the essay title "Mathematics is useful for science: discuss," you would be expected to comment on whether or not you think mathematics is useful for science and give your reasons, but you may take a position either way.
dummy variable: A synonym for "bound variable".
embedding: A map or function taking a structure $A$ (such as a group, ring, field, etc.) into another similar structure $B$ , so that the image of $A$ (considered as a substructure of $B$ ) looks exactly the same as $A$ . Such a function will always be injective and preserve any operations and relations present.
extension: Commonly a thing that makes a previous object bigger in some way. However this word is also used (especially in philosophy and logic) to mean "the actual object being defined" as opposed to the details of the specific "definition" that was used. Most mathematics is traditionally done as if the definitions of objects do not matter and only what the objects are matters: thus normal mathematics is "purely existensional". See also "intension".
identification: When two structures look identical (such as a structure and the image of it via an embedding) it often makes sense to regard the two structures as really being the same. We say that we identify them. Such identifications are not strictly logically correct, because the two copies of the same object really are different copies, but the simplification gained is almost always worthwhile. Examples include identifying the integers with the copy of the integers in a particular ordered field.
injection: A function $f : A \to B$ such that $f (x) \neq f (y)$ for all $x \neq y$ from $A$ . Injections are also called one-to-one functions.
intension: (Nb. this is not the same as "intention".) The intension of an object is the precise detail of how it is defined. Thus the sets $A = \{x \in ℕ: x^{2} - 5 x + 6 = 0\}$ and $B = \{2, 3\}$ have different intensions, but they happen to be the same extensionally. (See also "extension".)
isomorphism: A map or function taking a structure $A$ (such as a group, ring, field, etc.) exactly onto another structure $B$ of the same kind, so that both $A$ and $B$ look exactly the same. In other words, an isomorphism is an embedding that is surjective as well as injective.
joke: A light hearted interjection intended to cause amusement. See also synonymous.
line, line segment: Traditionally, a line is infinite in both directions. A part of a line, with one or two end points is called a line segment.
modus ponens: A Latin name for the rule of deduction that says "from A and A implies B deduce B".
necessarily: An adverb use to emphasise that some truth or implication is particularly strong. ("If $x = 2$ it necessarily follows that $x^{2} = x + x$ .") There is no good mathematical definition of the word and it is usually just used for emphasis, but some people have suggested that "necessary" could mean the same as "in all possible worlds/universes" (e.g. imagined ones as well as the real one) or it means "provable is some strong sense" (e.g. logically or mathematically).
predicate: A word from philosophical logic meaning a symbol (or phrase) for a property that an object may (or may not) have. The notation is usually $P x$ meaning the object $x$ has the property represented by the predicate $P$ .
is proportional to: Given two varying quantities $y$ and $x$ , $y$ is proportional to $x$ if there is a constant $C$ so that $y = C x$ . We also say, $y$ is inversely proportional to $x$ if there is a constant $C$ so that $y = C / x$ . (Usually the full situation under consideration is more complicated, and this law of proportionality requires other quantities being held constant. For example, for a gas, pressure is proportional to temperature at constant pressure.)
succedent: For a statement that is an implication, this denotes the part of that statement to the right of the word or symbol for "implies".
surjection: A function $f : A \to B$ such that for all $y \in B$ there is $x \in A$ with $y = f (x)$ . Surjections are also called onto functions.
synonymous: A synonym for "meaning the same as". See also "joke".
thesis: A statement to be argued. Typically used in mathematics to mean a statement that needs a careful argument but which by its nature cannot be given a complete water-tight rigorous mathematical proof.