/Solutions

1. Non-decreasing functions

Let A and B be partially-ordered sets. Recall that a function f:A→B is non-decreasing if x≤y implies f(x)≤f(y) for all x, y in A.

1. Prove or disprove: The set S₂ of all non-decreasing functions from ℕ to {0,1} is countable.

2. Prove or disprove: The set S_ℕ of all non-decreasing functions from ℕ to ℕ is countable.

(Assume the usual order on ℕ and {0,1}.)

2. Lex and colex orders

Let A and B be totally ordered sets. Here are three partial orders on A×B:

Lexicographic order

(a,b) ≤_L (a',b') iff a < a' or a = a' and b ≤ b'.

Colexicographic order

(a,b) ≤_C (a',b') iff b < b' or b = b' and a ≤ a'.

Product order

(a,b) ≤_× (a',b') iff a ≤ a' and b ≤ b'.

Prove or disprove: For any totally ordered sets A and B, the product order (≤_×) = (≤_L) ∩ (≤_C).

3. Addition and negation

Suppose we have an addition operation on some set S (i.e., a function +:S×S→S, written in infix form between its arguments), and we know that addition satisfies the axioms

• x+y=y+x (commutativity),
• x+(y+z) = (x+y)+z (associativity), and
• x+z = y+z ⇒ x=y (cancellation),

where x, y, z, are any elements of S.

Define the relation ~ on S×S by the rule (x,y) ~ (x',y') iff x+y' = x'+y.

1. Show that ~ is an equivalence relation.
2. Define the operation (x,y) + (x',y') = (x+x', y+y'). Show that if (x,y) ~ (z,z), then (x,y) + (x',y') ~ (x',y').
3. Define -(x,y) = (y,x). Show that (x,y) + (x',y') + -(x',y') ~ (x,y).