1.7 Relations
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 it was under 300 pages, whereas it would be quite acceptable for a math doctoral student to submit a thesis amounting to fewer 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 “\(2 < 3\)” but it’s not okay to write “\(3 < 2\).” The symbol \(<\) 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: \[ =, <, >, \leq, \geq,\; \!\mid\!\; , \; \mbox{and} \; \equiv \pmod{m}. \]
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).
What about expressions involving variables and these relation symbols? For example, what does \(x < y\) really mean? Okay, I know that you know what \(x < y\) 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 \(x < y\), \(x\) and \(y\) are just two arbitrary 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, \(<\). As a set of ordered pairs, this relation is the set \[ \{(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5) \}. \]
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 as shown in Figure 1.6.
Figure 1.6: A “betweenness” ternary relation.
The betweenness relation on the points in Figure 1.6 consists of the following triples: \[\begin{gather*} \{ (A,B,C), (A,G,D), (A,F,E), (B,G,E), (C,B,A), (C,G,F), (C,D,E), \\ (D,G,A), (E,D,C), (E,G,B), (E,F,A), (F,G,C) \}. \end{gather*}\]
Exercise 1.4 When thinking of a function as a special type of relation, the pairs are of the form \((x, f(x)).\) That is, they consist of an input and the corresponding output. What is the restriction that must be placed on the pairs in a relation if it is to be a function? (Hint: think about the so-called vertical line test.)
1.7.1 Exercises
Consider the numbers from 1 to 10. Give the set of pairs of these numbers that corresponds to the divisibility relation.
The domain of a function (or binary relation) is the set of numbers appearing in the first coordinate. The range of a function (or binary relation) is the set of numbers appearing in the second coordinate.
Consider the set \(\{0,1,2,3,4,5,6\}\) and the function \(f(x) = x^2 \pmod{7}\). Express this function as a relation by explicitly writing out the set of ordered pairs it contains. What is the range of this function?
What relation on the numbers from 1 to 10 does the following set of ordered pairs represent? \[\begin{gather*} \{ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (1,10), \\ (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,8), (2,9), (2,10), \\ (3,3), (3,4), (3,5), (3,6), (3,7), (3,8), (3,9), (3,10), \\ (4,4), (4,5), (4,6), (4,7), (4,8), (4,9), (4,10), \\ (5,5), (5,6), (5,7), (5,8), (5,9), (5,10), \\ (6,6), (6,7), (6,8), (6,9), (6,10), \\ (7,7), (7,8), (7,9), (7,10), \\ (8,8), (8,9), (8,10), \\ (9,9), (9,10), \\ (10,10) \} \end{gather*}\]
Draw a five-pointed star and label all 10 points. There are 40 triples of these labels that satisfy the betweenness relation. List them.
Sketch a graph of the relation \[ \{ (x,y) {\,:\,}x,y \in {\mathbb R}\; \mbox{and} \; y > x^2 \}. \]
A function \(f(x)\) is said to be invertible if there is another function \(g(x)\) such that \(g(f(x)) = x\) for all values of \(x\). (Usually, the inverse function, \(g(x)\) would be denoted \(f^{-1}(x)\).) Suppose that a function is presented to you as a relation — that is, you are just given a set of pairs. How can you distinguish whether the function represented by this list of input/output pairs is invertible? How can you produce the inverse (as a set of ordered pairs)?
There is a relation known as “has colour” which goes from the set \[ F = \{\text{orange}, \text{cherry}, \text{pumpkin}, \text{banana}\} \] to the set \[ C = \{\text{orange}, \text{red}, \text{green}, \text{yellow}\}. \]
What pairs are in “has colour”?