2.2 Some Examples

`typewriter font`

. Maple's responses are shown in

> 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).

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

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.

Hence, for example

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

Multiplication table for

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.

- 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.

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.