Then we tell Simon that we shall mark with 1 the cases where a is greater than b and mark with 0 the cases where a is not greater than b:
| Greater Than | |||||
|---|---|---|---|---|---|
| b: | 0 | 1 | 2 | 3 | |
| a: | |||||
| 0 | 0 | 0 | 0 | 0 | |
| 1 | 1 | 0 | 0 | 0 | |
| 2 | 1 | 1 | 0 | 0 | |
| 3 | 1 | 1 | 1 | 0 | |
For example, “2 is greater than 3” is false, so we put 0 in the table on the 2 line in the 3 column. We see that, for the 16 possible cases, a is greater than b in 6 cases and a is not greater than b in 10 cases.
There is a neat way of saying what we have just said, using the language of mathematical logic (see [Chapter 9] and [Supplement 2]). Suppose that we consider the statement “a is greater than b” where a and b may be any of the numbers 0, 1, 2, 3. We can say that the truth value p of a statement P is 1 if the statement is true and that it is 0 if the statement is false:
p = 1 if P is true, 0 if P is false
The truth value of a statement P is conveniently denoted as T(P) ([see Supplement 2]):
p = T(P)
Now we can say that the table for the operation greater than shows the truth value of the statement “a is greater than b”:
p = T(a > b)
Let us turn now to the operation selection. By selecting we mean choosing one number a if some statement P is true and choosing another number b if that statement is not true. As before, let p be the truth value of that statement P, and let it be equal to 1 if P is true and to 0 if P is false. Then the operation of selection is fully expressed in the following table and logical formula ([see Supplement 2]):