1. Uncountable sets & fixed points
CS 350 — Fall 2018 gilray.org/classes/fall2018/cs350/

2. 𝐹:ℕ→ℙ
𝐹(𝑛)=𝑛+1 1 2 3 4 5 6 7 8 9 10 11
If a bijective function (injective, surjective) exists between two finite
sets, then they must have the same cardinality. The same reasoning
may be extended to infinite sets.

3. Cantor’s Diagonalization Argument

4. There exists no surjective function from the natural numbers to the set of real numbers in just the range [0.0, 1.0]

5. … = …

6. n f(n) … … … … … … … … …

7. n f(n) … … … … … … … … …
d …

8. n f(n) … … … … … … … … …
d …
d’ cannot exist as it must
disagree at the d’th digit!
d’ …

9. Fixed-point algorithms

10. What are fixed points
(a.k.a. fixpoints)?

11. f(x) = x2
(1,1)
(0,0)

12. {𝑥|(𝑥,𝑥)∈𝐹} or
{𝑥|𝐹(𝑥)=𝑥}

13. Babylonian method for computing .2

14. 𝑥 2 =𝑁
𝑥= 𝑁 𝑥

15. 𝑥 2 =𝑁
𝑓(𝑥)= 𝑁 𝑥

16. 𝑥 2 =𝑁
𝑓(𝑥)= 𝑁 2𝑥 + 𝑥 2

17. Requirements for fixed-point iteration:

18. 𝑓:𝐷→𝐷 𝑓(𝑥) 𝑓(𝑥)=𝑓(𝑓(𝑥)) 𝑥 1. A function with a (least) fixed-point.
2. Progress: is a strictly better estimate than , or it is a fixed-point:
𝑓(𝑥)
𝑓(𝑥)=𝑓(𝑓(𝑥)) 𝑥
(Often by showing monotonicity.)

19. Monotone/Monotonic functions preserve order, thus:
Monotonicity
𝑓is monotonic iff∀𝑥,𝑦.𝑥≥𝑦⟹𝑓(𝑥)≥𝑓(𝑦)
Monotone/Monotonic functions preserve order, thus:
𝑓(𝑥)≥𝑥⟹𝑓(𝑓(𝑥))≥𝑓(𝑥)

20. 𝑓 𝐷 𝑓:𝐷→𝐷 𝑓(𝑥) 𝑓(𝑥)=𝑓(𝑓(𝑥)) 𝑥
1. A function with a (least) fixed-point.
𝑓:𝐷→𝐷
2. Progress: is a strictly better estimate than , or it is a fixed-point:
𝑓(𝑥)
𝑓(𝑥)=𝑓(𝑓(𝑥)) 𝑥
(Often by showing monotonicity.)
3. Bounded number of iterations possible.
(Often by showing continuous and finite ) 𝑓 𝐷

21. Computing transitive closure.

22. (𝑥,𝑦)∈𝑅∧(𝑦,𝑧)∈𝑅⟹(𝑥,𝑧)∈𝑅
𝑓(𝑅)=𝑅∪{(𝑥,𝑧)|(𝑥,𝑦)∈𝑅∧(𝑦,𝑧)∈𝑅}