1. Sets and Logic…. Chapter 3
An experiment…..
Sets and Logic…. Chapter 3

2. Logical Laws DeMorgan’s Laws Commutative, Associative, and Idempotent
¬ 𝑃∧𝑄 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 ¬𝑃∨¬𝑄
¬ 𝑃∨𝑄 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 ¬𝑃∧¬𝑄
Commutative, Associative, and Idempotent
Distributive
𝑃∧ 𝑄∨𝑅 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 (𝑃∧𝑄)∨(𝑃∧𝑅)
𝑃∨ Q∧𝑅 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 𝑃∨𝑄 ∧ 𝑃∨𝑅
Absorption: 𝑃∨ 𝑃∧𝑅 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 𝑃
𝑃∧ 𝑃∨𝑅 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 𝑃

3. Equivalences
𝑃→𝑄 is equivalent to ¬𝑃∨𝑄
𝑃→𝑄 is equivalent to ¬(𝑃∧¬𝑄)

4. The Converse and Contrapositive
Consider 𝑃→𝑄 and 𝑄→𝑃
𝑄→𝑃 is called the converse of 𝑃→𝑄
¬𝑄→¬𝑃 is the contrapositive of 𝑃→𝑄
The converse of 𝑃→𝑄 is not equivalent
The contrapositive is……..
𝑃→𝑄 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 ¬𝑄→¬𝑃

5. Chapter 2: Quantifiers Three ideas
For all or for every, we use the notation: ∀
There is one, or there exists: ∃
The universe……

6. Seven Operators Connectives Quantifiers
¬ ∧ ∨ → ↔
Quantifiers
∀ 𝑎𝑛𝑑 ∃
This is really all we need to write any mathematical statement!

7. New Notation
∀𝑥 𝑥>2→ 𝑥 2 >4
Is there and implied universe?

8. EquivaleNce Involving Quantifiers
Quantifier Negation Laws
¬∃𝑥𝑃 𝑥 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜∀𝑥¬𝑃 𝑥
¬∀𝑥𝑃 𝑥 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 ∃𝑥¬𝑃(𝑥)

9. Chapter 3: Proofs
We want to build a “cookbook” of strategies for proving different relations
The goal is to be systematic, and flexible
Different strategies work with different situations

10. 3.1 Proof Strategies Givens and Goals
Often times theorems state their givens, i.e. defining the universe, etc. and finish with simple statements stated as 𝑃→𝑄.
Givens: Goal:
Prove 𝑃→𝑄

11. 3.1 Proof Strategies Move P to the givens Givens Goal -------------- 𝑄𝑄 P

12. Example
Suppose a and b are real numbers. If 0<𝑎<𝑏 then 𝑎 2 < 𝑏 2 .
Givens: Goal:
a and b are real numbers 𝑃→𝑄
a and b are real numbers 𝑄
P or 0<𝑎<𝑏

13. Form of Basic Proof 𝑃→𝑄 Givens Goal Final Form ------------- Q
Suppose P and other Givens
Prove Q
Thus 𝑃→𝑄

14. Alternate Strategy: Contrapositive
𝑃→𝑄 is equivalent to ¬𝑄→¬𝑃
Givens Goal Q P ¬𝑃 ¬𝑄

15. Contrapositive proof
Assume givens and ¬𝑄
Prove ¬𝑃
Therefore 𝑃→𝑄

16. Example
Suppose a, b, and c are real numbers and a>b. If 𝑎𝑐≤𝑏𝑐 then 𝑐≤0.
Givens Goal
a,b,c, are real numbers (𝑎𝑐≤𝑏𝑐)→(𝑐≤0)
a>b
a,b,c, are real numbers 𝑎𝑐>𝑏𝑐
c>0

17. Proofs involving Negations and Conditionals
Proving something of the form ¬𝑃
Try to reexpress ¬𝑃 positively
Example: Suppose 𝐴∩𝐶⊆𝐵 and 𝑎∈𝐶. Prove that 𝑎∉𝐴\B

18. Additional Possibility
Givens Goal ¬𝑃
Assume P and look for a contradiction
Contradiction P

19. Example If 𝑥 2 +𝑦=13, and 𝑦≠4 then 𝑥≠3.
Prove it two ways… or set it up two ways

20. Plug and Chug… write the definitions out
Suppose A,B, and C are sets, 𝐴\B⊆𝐶, and x is anything at all. Prove that if 𝑥∈𝐴\C, then 𝑥∈𝐵.
Try proceeding by contradiction

21. Silly Latin Modus ponens Modus tollens 𝑃→𝑄 and 𝑃 implies 𝑄

22. Example
Suppose 𝑃→ 𝑄→𝑅 , then ¬𝑅→(𝑃→¬𝑄).

23. Example
Suppose that 𝐴⊆𝐵, 𝑎∈𝐴, and 𝑎∉𝐵\C. Then 𝑎∈𝐶.

24. Proofs involving quantifiers
∀𝑥𝑃 𝑥
Instead let x be arbitrary
Suppose A, B, and C are sets, and 𝐴\B⊆𝐶. Prove that 𝐴\C⊆𝐵

25. Example
Suppose A and B are sets. Prove that if 𝐴∩𝐵=𝐴, then 𝐴⊆𝐵.

26. Existence Proofs ∃𝑥𝑃 𝑥 You only have to find one x
Sometimes by example
Often times done by contradiction

27. Existence Proof ( Simple )
Prove that for every real number x>0, there is a real number y such that 𝑦 𝑦−1 =𝑥.

28. Example from Analysis

29. Example
Suppose that ℱ 𝑎𝑛𝑑 ℋ are families of sets and that ℱ∩ℋ≠∅. Prove that ∩ℱ⊆∪ℋ.
Make sure to write the definitions first.

30. Example
Suppose B is a set and ℱ is a family of sets. Prove that if ∪ℱ⊆𝐵 then ℱ⊆℘ 𝐵 .
Write out the definitions first

31. Example
Theorem: For all integers a,b, and c, if a|b and b|c then a|c.

32. 3.4: Proofs with ∧,∨, … ↔ To Prove a Goal of 𝑃∧𝑄 To Use a Given of 𝑃∧𝑄
Prove them separately
To Use a Given of 𝑃∧𝑄
Treat them separately
Example: Suppose 𝐴⊆𝐵, and A and C are disjoint. Prove that 𝐴⊆𝐵\C.

33. Biconditionals: ↔ To prove a goal of 𝑃↔𝑄
Prove 𝑃→𝑄 and 𝑄→𝑃 separately
To use a given of the form 𝑃↔𝑄
Treat 𝑃→𝑄 and 𝑄→𝑃 separately
Example: Suppose x is an integer. Prove that x is even if and only if 𝑥 2 is an integer.

34. Example Suppose A, B, and C are sets. Prove that 𝐴∩ 𝐵\C =(𝐴∩𝐵)\C

35. Example
For every integer n, 6|n iff 2|n and 3|n.

36. Proof with disjunctions, ∨
To prove (𝑃∨𝑄)→𝑅
Prove 𝑃→𝑅
And
Prove 𝑄→𝑅

37. Examples
Suppose A,B, and C are sets. Prove that if 𝐴⊆𝐶 𝑎𝑛𝑑 𝐵⊆𝐶 𝑡ℎ𝑒𝑛 𝐴⋃𝐵⊆𝐶.
𝐴\(𝐵\C)⊆(𝐴\B)⋃𝐶

38. Example
For every integer x, the remainder when 𝑥 2 is divided by 4 is either 0 or 1.
For every real number x, if 𝑥 2 ≥𝑥 then either 𝑥≤0 𝑜𝑟 𝑥>1.

39. Example
Suppose m and n are integers. If mn is even then either m is even or n is even.