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2.2 Some Examples

2.34   Example (non-commutative .) Many computer languages support an and operation that is not commutative. Here is a script of a Maple session. My statements are shown in typewriter font. Maple's responses are shown in italics.

> P := (x = 1/y);


> Q := (x*y=1);


> y:= 0;


> x := 1;


> Q and P;


> P and Q;


Error, division by zero

When evaluating , Maple first found that is false, and then, without looking at , concluded that must be false. When evaluating , Maple first tried to evaluate , and in the process discovered that is not a proposition. Mathematically, both and are errors when and . Many programmers consider the non-commutativity of to be a feature (i.e. good), rather than a bug (i.e. bad).

2.35   Example (Calculator operations.) Let denote the set of all numbers that can be entered into your calculator. The exact composition of depends on the model of your calculator. Let where is some object not in . I will call the error. I think of as the result produced when you enter . Define four binary operations , and on by

On my calculator

If denotes any of , I define

On all calculators with which I am familiar, and are commutative operations, is an identity for , is an identity for , and every element of except for is invertible for . On my calculator
 (2.36) (2.37) (2.38) (2.39)

Thus has two different inverses! It follows from theorem 2.15 that is not associative. Your calculator may give different results for the calculations (2.38) and (2.39) but none of the calculator operations are associative.

2.40   Exercise. A Verify that calculator addition and calculator multiplication are not associative, by finding calculator numbers , , , , , and such that , and .

2.41   Notation. If , let

Hence, for example

2.42   Definition ( .) Let , with . We define two binary operations and on by:

for all ,

and for all ,

Thus,

and

The operations and are both commutative (since and are commutative on ). Clearly is an identity for , and is an identity for . Every element of is invertible for and

2.43   Definition (Multiplication table.) Let be a binary operation on a finite set having elements. We construct a multiplication table for as follows: We write down a table with rows and columns. Along the top of the table we list the elements of as labels for the columns. Along the left side of the table we list the elements of (in the same order) as labels for the rows. (See the figure to see what is meant by this.) If , we write the product in the box of our table whose row label is and whose column label is .

Multiplication table for

2.44   Examples. Below are the multiplication tables for and :

 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3

 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1

By looking at the multiplication table for we see that

Hence all the non-zero elements of have inverses under .

Both of the operations and are associative. This follows from the fact that and are associative operations on , by a straightforward but lengthy argument. The details are given in appendix B.

2.45   Exercise. Write down the multiplication table for on . Determine which elements of are invertible for , and find the inverse for each invertible element.

2.46   Exercise. Let be a set containing three distinct elements. , , . Let be the binary operation on determined by the multiplication table:

a)
Show that there is an identity element for . (Which of is the identity?)
b)
Show that has two different inverses for .
c)
Explain why the result of part b does not contradict the theorem on uniqueness of inverses.

2.47   Note. An early example of a binary operation that was not an obvious generalization of one of the operations on numbers was the use of union and intersection as binary operations on the set of all sets by George Boole[11]. In Laws of Thought (1854), Boole introduces the operation (for union) and (for intersection) on  classes" (although he usually writes instead of ). He explicitly states

which he calls commutative and distributive laws. He does not mention associativity, and writes without parentheses. He denotes  Nothing" by and  the Universe" by , and notes that and have the usual properties. As an example of the distributive law, Boole gives

European men and women European men and European women.

Boole's is not really a binary operation since he only defines when and have no elements in common.

The word associative, in its mathematical sense, was introduced by William Hamilton[24, p114] in 1843 in a paper on quaternions. According to [14, p284], the words commutative and distributive were introduced by Francois -Joseph Servois in 1813.

Next: 2.3 The Field Axioms Up: 2. Fields Previous: 2.1 Binary Operations   Index