One of the principle ways in which mathematical writing differs from ordinary writing is in its incredible brevity. For instance, a Ph.D. thesis for someone in the humanities would be very suspicious if its length were less than 300 pages, whereas it would be quite acceptable for a math doctoral student to submit a thesis amounting to less than 100 pages. Indeed, the usual criteria for a doctoral thesis (or indeed any scholarly work in mathematics) is that it be “new, true and interesting.” If one can prove a truly interesting, novel result in a single page — they'll probably hand over the sheepskin.

How is this great brevity achieved? By inserting single symbols in place of a whole paragraph's worth of words! One class of symbols in particular has immense power — so-called relational symbols. When you place a relational symbol between two expressions, you create a sentence that says the relation holds. The period at the end of the last sentence should probably be pronounced! “The relation holds, period!” In other words when you write down a mathematical sentence involving a relation, you are asserting the relation is True (the capital T is intentional). This is why it's okay to write “\(2 \lt 3\)” but it's not okay to write “\(3 \lt 2\).” The symbol \(\lt\) is a relation symbol and you are only supposed to put it between two things when they actually bear this relation to one another.

The situation becomes slightly more complicated when we have variables in relational expressions, but before we proceed to consider that complication let's make a list of the relations we've seen to date: \begin{equation*} =, \lt , >, \leq, \geq,\; \divides \; , \; \mbox{and} \; \equiv \pmod{m}. \end{equation*}

Each of these, when placed between numbers, produces a statement that is either true or false. Ordinarily we wouldn't write down the false ones, instead we should express that we know the relation doesn't hold by negating the relation symbol (often by drawing a slash through it, but some of the symbols above are negations of others).

So what about expressions involving variables and these relation symbols? For example what does \(x \lt y\) really mean? Okay, I know that you know what \(x \lt y\) means but, philosophically, a relation symbol involving variables is doing something that you may have only been vaguely aware of in the past — it is introducing a supposition. Watch out for relation symbols involving variables! Whenever you encounter them it means the rules of the game are being subtly altered — up until the point where you see \(x \lt y\), \(x\) and \(y\) are just two random numbers, but after that point we must suppose that \(x\) is the smaller of the two.

The relations we've discussed so far are binary relations, that is, they go in between two numbers. There are also higher order relations. For example, a famous ternary relation (a relationship between three things) is the notion of “betweenness.” If \(A\), \(B\) and \(C\) are three points which all lie on a single line, we write \(A\star B \star C\) if \(B\) falls somewhere on the line segment \(\overline{AC}\). So the symbol \(A\star B \star C\) is shorthand for the sentence “Point \(B\) lies somewhere in between points \(A\) and \(C\) on the line determined by them.”

There is a slightly silly tendency these days to define functions as being a special class of relations. (This is slightly silly not because it's wrong — indeed, functions are a special type of relation — but because it's the least intuitive approach possible, and it is usually foisted-off on middle or high school students.) When this approach is taken, we first define a relation to be any set of ordered pairs and then state a restriction on the ordered pairs that may be in a relation if it is to be a function. Clearly what these Algebra textbook authors are talking about are binary relations, a ternary relation would actually be a set of ordered triples, and higher order relations might involve ordered 4-tuples or 5-tuples, etc. A couple of small examples should help to clear up this connection between a relation symbol and some set of tuples.

Consider the numbers from 1 to 5 and the less-than relation, \(\lt\). As a set of ordered pairs, this relation is the set \begin{equation*} \{(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5) \}. \end{equation*}

The pairs that are in the relation are those such that the first is smaller than the second.

An example involving the ternary relation “betweenness” can be had from the following diagram.

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The betweenness relation on the points in this diagram consists of the following triples. \begin{gather*} \{ (A,B,C), (A,G,D), (A,F,E), (B,G,E), (C,B,A), (C,G,F), (C,D,E),\\ (D,G,A), (E,D,C), (E,G,B), (E,F,A), (F,G,C) \}. \end{gather*}