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.
Prove or disprove: The set S2 of all non-decreasing functions from ℕ to {0,1} is countable.
Prove or disprove: The set Sℕ 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.
- Show that ~ is an equivalence relation.
- 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').
- Define -(x,y) = (y,x). Show that (x,y) + (x',y') + -(x',y') ~ (x,y).