In the previous post of this series, we looked more closely at the mathematical notion of equivalence, through the lens of operators. Specifically, we looked at a split (pair) 2|13 and then looked at an operator that turns 2|13 into 3|12, turns 3|12 into 4|11, turns 4|11 into 5|10, etc., all leaving the total the same (15). We said that the splits 2|13, 3|12, 4|11, 5|10 are all equivalent with respect to the + operator.
Let’s take another look at equivalence with respect to the + operator, by examining an addition table. An addition table is similar to the more familiar multiplication table – works the same way, serves the same function – even though most of us never encountered one in grade school. In fact, of the four operations +, -, ×, ÷, only multiplication seems to have been awarded its special table. Have you ever wondered why that is?
The figure on the left shows an addition table, though you may have expected us to start at 1 and end at 10. Instead, we started at 0 and ended at 13, and we could make the table as large as we wanted. To add 8 + 9, using the table, we would look at the row labeled “8” and the column labeled “9”, and then follow that row and that column till you find the place where they meet. The number you find there is 17 – the sum of 8 and 9.
There are a lot of numbers in this addition table, but then many of them are the same. The pattern of the numbers is not hard to pick up. The figure on the right shows this by drawing blue lines that follow sums that are the same. In the table on the right, to add 8 + 9, we would again look for the position in the table that’s in row 8 and column 9. We would then follow the blue arrow to the number at its end, and read off the result: 17.
Each blue line, together with the numbers at either end, marks a set of equivalent splits (equivalent with respect to the + operator). Same blue line, same sum. Different blue line, different sum. Each blue line marks an equivalence class, that is, a collection of items that are all equivalent to each other. One of the items in the equivalence class may be designated as the representative of that class, the one that is more equal than the others. In our example, the representative item of the equivalence class for the sum of 17 might be the split at the head of the arrow, 4|13. If we had drawn a bigger addition table, we might have picked 0|17 as the representative of that same equivalence class. From one perspective, it doesn’t matter one iota which element of the equivalence class is picked to be the representative. Among those elements that are equivalent, we are free to pick one as being “more equal than the others”. In our diagram (the one on the right), each blue line marks an equivalence class, and the arrow on the blue line suggests how to find the representative of the equivalence class.
What we did with the addition table could also be done using multiplication tables. This is an idea we will pursue in the next post in this series.