The concept of transitivity goes back at least 2,300 years. In the Elements, Euclid includes it as one of his "common notions." He says, "Things which are equal to the same thing are also equal to one another." As Euclid puts it, if a = b and c = b, then a = c, which is equivalent to the modern version, which has "b = c" rather than "c = b."
Transitivity is a property of any relation between numbers, geometric figures, or other mathematical elements. A relation R is said to be transitive if a R b and b R c imply that a R c. For example, 6/4 = 3/2 and 3/2 = 1.5, therefore 6/4 = 1.5.
Of course, one would not be likely to make use of the transitive property to establish such an obvious fact, but there are cases where the transitive property is very useful. If one were given the two equations
one could use transitivity (after squaring both sides of the second equation) to eliminate x.
Transitivity is one of three properties which together make up an "equivalence relation."
|Transitive law||If a R b and b R c, then a R c|
|Reflexive law||a R a|
|Symmetric law||If a R b, then b R a|
To be an equivalence relation R must obey all three laws.
A particularly interesting relation is "wins over" in the game scissors-paper-rock. If a player chooses "paper," he or she wins over "rock;" and if the player chooses "rock," that wins over "scissors;" but "paper" does not win over "scissors." In fact, it loses. Although the various choices are placed in a winning-losing order, it is a non-transitive game. If it were transitive, of course, no one would play it.
In Wheels, Life, and Other Mathematical Amusements, Gardner describes a set of non-transitive dice. Die A has its faces marked 0, 0, 4, 4, 4, and 4. Each face of die B is marked with a 3. Die C is marked 2, 2, 2, 2, 6, 6. Die D is marked 1, 1, 1, 5, 5, 5. Each player chooses a die and rolls it. The player with the higher number wins. The probability that A will win over B is 2/3. The probability that B will win over C or that C will win over D is also 2/3. And, paradoxically, the probability that D will win over A is also 2/3. Regardless of the die which the first player chooses, the second player can choose one which gives him a better chance of winning. He need only pick the next die in the unending sequence... < A < B < C < D < A < B <...
There are many relations in life which are theoretically transitive, but which in practice are not. One such is the relation "like better than." One can like apples better than bananas because they are juicier, pomegranates better than apples because they have more flavor, and still like bananas better than pomegranates. They are, after all, a lot easier to eat. Transitivity, reflexivity, and symmetry are properties of very simple, one-dimensional relations such as one finds in mathematics but not in much of ordinary life.
Birkhoff, Garrett, and MacLane, Saunders. A Survey of Modern Algebra. New York: The Macmillan Co. 1947.
Dantzig, Tobias. Number, the Language of Science. Garden City, N. Y.: Doubleday and Co., 1954.
Gardner, Martin. Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman and Co., 1983.
J. Paul Moulton
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