1. Discrete Math & Fixed points
CS 350 — Fall 2018 gilray.org/classes/fall2018/cs350/

2. 𝐴={1,2,3}

3. 𝐴×𝐴=
{(1,1),(1,2),(1,3),
(2,1),(2,2),(2,3),
(3,1),(3,2),(3,3)}

4. ℕ×ℕ= {(0,0),(0,1),(0,2),(0,3),⋯ (1,0),(1,1),(1,2),(1,3),⋯
(2,0),(2,1),(2,2),(3,3),⋯
(3,0),(3,1),(3,2),(3,3), ⋱ ⋮

5. 𝑅⊆ℕ×ℕ

6. ∃𝑥∈𝐴.Even(𝑥) ∀𝑥∈𝐴.Even(𝑥) 𝐴={1,2,3} Even(𝑥) = Δ 𝑥≡0(mod2)
Propositions are statements that are True or False. E.g.,
𝐴={1,2,3}
Predicates are functions that yield True or False. E.g.,
Even(𝑥) = Δ 𝑥≡0(mod2)
Quantifiers lift a proposition over elements to a proposition over sets. E.g.
∃𝑥∈𝐴.Even(𝑥)
∀𝑥∈𝐴.Even(𝑥)

7. Applies to sets as well:
De Morgan’s Laws
𝑎∨𝑏⟺¬(¬𝑎∧¬𝑏)
𝑎∧𝑏⟺¬(¬𝑎∨¬𝑏)
Applies to sets as well:
𝐴∩𝐵⟺ ( 𝐴 ∪ 𝐵 )

8. Or, using set-builder notation:
𝑅={(0,1),(1,2),(2,3),(3,4),…}
Or, using set-builder notation:
𝑅={(𝑛,𝑛+1)|𝑛∈ℕ}

9. Or, as a graph: 1 2 3 4 5

10. |𝐴|=3
|𝐴×𝐴|=|𝐴|×|𝐴|=9
|𝒫(𝐴)|=| 2 𝐴 |= 2 |𝐴| =8

11. Counting Problems

12. 𝒫(𝐴)= 2 𝐴 =
{Ø,{1},{2},{3},
{1,2},{1,3},{2,3},{1,2,3}}

13. 𝒫(𝐴)= 2 𝐴 =
“Hasse” diagram
{1,2,3}
{1,2}
{1,3}
{2,3}
{1}
{2}
{3} Ø

14. (⊆)⊆𝒫(𝐴)×𝒫(𝐴)

15. 𝑥𝑅𝑦⟹𝑦𝑅𝑥 𝑥𝑅𝑦∧𝑦𝑅𝑥⟹𝑥=𝑦 𝑥𝑅𝑦∧𝑦𝑅𝑧⟹𝑥𝑅𝑧 ∀𝑥.𝑥𝑅𝑥 Reflexivity of R:
Symmetry of R:
𝑥𝑅𝑦⟹𝑦𝑅𝑥
Antisymmetry of R:
𝑥𝑅𝑦∧𝑦𝑅𝑥⟹𝑥=𝑦
Transitivity of R:
𝑥𝑅𝑦∧𝑦𝑅𝑧⟹𝑥𝑅𝑧

16. ⊑ ⊑ (ℝ,≤) (𝒫(ℕ),⊆) ∀𝑥,𝑦.𝑥𝑅𝑦∨𝑦𝑅𝑥
Partial orders are reflexive, antisymmetric, and transitive.
If , then the order is total.
If it is not antisymmetric, then it’s a preorder.
If it is irreflexive, then it is a strict partial order.
A set S and a partial order ( ) form a poset (S, ) together—a partially ordered set.
For example: or
A equivalence relation is reflexive, symmetric, and transitive—has equivalence classes. What does it’s graph look like?
∀𝑥,𝑦.𝑥𝑅𝑦∨𝑦𝑅𝑥 ⊑ ⊑
(ℝ,≤)
(𝒫(ℕ),⊆)

17. A join , also called a least upper bound, is the unique element of a poset no less than than x and y, but strictly less than all other elements no less than either x or y.
A meet , also called a greatest lower bound, is the unique element of a poset no greater than than x and y, but strictly greater than all other elements no greater than either x or y.
A lattice is a partially ordered set such that, for all x and y, both a meet and a join exist for x and y.
(𝑥⊔𝑦)
𝑥⊑(𝑥⊔𝑦)∧𝑦⊑(𝑥⊔𝑦)∧∀𝑧.(𝑥⊑𝑧∧𝑦⊑𝑧∧𝑧⊑(𝑥⊔𝑦)⟹𝑧=(𝑥⊔𝑦)
(𝑥⊓𝑦)
𝑥⊒(𝑥⊓𝑦)∧𝑦⊒(𝑥⊓𝑦)∧∀𝑧.(𝑥⊒𝑧∧𝑦⊒𝑧∧𝑧⊒(𝑥⊓𝑦)⟹𝑧=(𝑥⊓𝑦)

18. (1.0m, 100kg)
(0.5m, 100kg)
(1.0m, 50kg)
(1.0m, 10kg)
(0.5m, 50kg)
(0.5m, 10kg)

19. ∀𝑥.∃𝑦.(𝑥,𝑦)∈𝐹 ∀𝑦.∃𝑥.(𝑥,𝑦)∈𝐹 ∀𝑥,𝑦.𝐹(𝑥)=𝐹(𝑦)⟹𝑥=𝑦
F is a function:
∀𝑥,𝑦,𝑧.(𝑥,𝑦)∈𝐹∧(𝑥,𝑧)∈𝐹⟹𝑦=𝑧
F is a total function:
∀𝑥.∃𝑦.(𝑥,𝑦)∈𝐹
F is injective/1-1:
∀𝑥,𝑦.𝐹(𝑥)=𝐹(𝑦)⟹𝑥=𝑦
F is surjective/onto:
∀𝑦.∃𝑥.(𝑥,𝑦)∈𝐹

20. sets, then they must have the same cardinality.
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.

21. n f(n) … … … … … … … … …

22. n f(n) … … … … … … … … …
d …

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

24. Cantor’s Diagonalization Proof

25. Fixed-point algorithms

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

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

28. {𝑥|(𝑥,𝑥)∈𝐹} or
{𝑥|𝐹(𝑥)=𝑥}

29. Babylonian method for computing .2