ABSTRACT
Preparation for Chapter 6
Prep Problem 6.1. One can define the partial order on positive integers where a ≤ b if a exactly divides b. Here is a diagram (called a Hasse diagram) of the factors of 12 under this partial order: https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781482243048/27066273-287e-4f87-9432-f815edbc9855/content/pg138_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Note that in the diagram, a line is drawn from a upward toward b if a < b and no other c fits in between (i.e., any comparisons can be inferred from the diagram).
Draw two Hasse diagrams, one each for the factors of 36 and of 30.
How would you determine the greatest common divisor (gcd) and the least common multiple (lcm) of two elements by looking at Hasse diagrams?
Prep Problem 6.2.List all games each with a single left option and a single right option each chosen from {1, 0, −1}. Draw the partial order of these nine games. (The diagram should look like that of the factors of 36 from Prep Problem 6.1.)
How many additional games would there be if we allowed the left or right options to be empty? List them.
To the instructor: While we make use of thematerial in Section 6.3 in Chapter 8, the material in Sections 6.4 and 6.5 can be skipped without loss of continuity.
… a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
Godfrey Hardy in A Mathematician’s Apology