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# Transitive

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In mathematics (and especially within logic), the transitive property denotes a relationship between three elements in which: if the relationship is valid between the first and second elements and is valid between the second and third elements, then the same relationship must be valid between the first and third elements. Equal numbers are said to be transitive, as will be shown soon. The concept of transitivity goes back at least 2,300 years. In the Elements, Greek mathematician Euclid of Alexandra (c. 325c. 265 BC) 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

y = x 2

x = z + 1

one could use transitivity (after squaring both sides of the second equation) to eliminate x.

y = z 2+ 2z + 1

Transitivity is one of three properties that together make up an equivalence relation.

Transitive law

Reflexive

Symmetric law

If a R b and b R c, then a R c a

law R a

If a R b, than 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, which 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.

Transition metals see Periodic table

## KEY TERMS

Reflexive A relation R is reflexive if for all elements a, a R a.

Symmetric A relation R is symmetric if for all elements a and b, a R b implies that b R a.

Transitive A relation R is transitive if for all elements a, b, and c, a R b and b R c implies that a R c.

In Wheels, Life, and Other Mathematical Amusements, Martin 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, and 6. Die D is marked 1, 1, 1, 5, 5, and 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 that the first player chooses, the second player can choose one that gives him/her a better chance of winning. He/she need only pick the next die in the unending sequence < A< B< C< D< A< B<....

There are many relations in life that are theoretically transitive, but that 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.

## Resources

### BOOKS

Birkhoff, Garrett, and Saunders MacLane. A Survey of Modern Algebra. New York: Macmillan Co., 1947.

Burton, David M. Elementary Number Theory. Boston:McGraw -Hill Higher Education, 2007.

Dantzig, Tobias. Number, the Language of Science. Garden City, NY: Doubleday and Co., 1954.

Gardner, Martin. Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman and Co., 1983.

Lorenz, Falko. Algebra. New York: Springer, 2006. Setek, William M. Fundamentals of Mathematics. Upper

Saddle River, NJ: Pearson Prentice Hall, 2005.

J. Paul Moulton