Assignment 1, Math 3TP3
1. In class, we showed how Euclid constructed an equilateral triangle. Use this construction to construct a regular hexagon. Generalize this to construct a regular 3 · 2 n -gon.
2. Recall that in class we constructed sets An for every n such that if one derived them n times they remained non-empty but if you derived them n + 1 times. This led to the construction of a set Aω with the property that A(n) ω was non-empty for every n but T n A(n) ω = ∅. We will now construct a set A such that you have to derived A ω + 1-many times before the set disappears. That is, B = \ n A (n) 6= ∅ but B 0 = ∅.
Hint: Take the example Aω we constructed in class as mentioned above. Now recall that tan−1 maps R to the bounded interval (−π/2, π/2). This allows you to squish even something constructed on all of R to an interval. Try to piece together squished copies of Aω to get the required set A.
3. Let’s create a dictionary between logical notions and characteristic functions: Suppose that A and B are subsets of N. Express, in terms of the characteristic functions χA and χB, the following:
(a) the characteristic function of N \ A i.e. the set of elements of N not in A.
(b) the characteristic function of A ∪ B.
(c) the characteristic function of A ∩ B.
(d) Suppose that A ⊂ N2 . How do you express the characteristic function of “there exists y such that (x, y) ∈ A”?
4. Show that for any set A, there is no surjective map from A to P(A). This shows that the cardinality of A is less than that of P(A) for every set A.