How To Approach A Mathematical Proofhttps://people.ucalgary.ca/~rscheidl/crypto/handouts/Tutorial/proof-approach.pdf
Proof Writing Tipshttps://people.ucalgary.ca/~rscheidl/crypto/handouts/Tutorial/Proof%20writing%20tips.pdf
Sample Proofshttps://people.ucalgary.ca/~rscheidl/crypto/handouts/Tutorial/proofs.pdf
The incidence of breast cancer is 1%. A specific test for breast cancer has a false negative rate of 10% (ie. it is correct 90% of the time when the patient has cancer) and a false positive rate of 9% (ie. it is correct 91% of the time when the patient doesn’t have cancer). A patient takes this test and gets a positive result. What are the odds they have breast cancer?
The incidence of breast cancer is 1%. A specific test for breast cancer has a false negative rate of 10% (ie. it is correct 90% of the time when the patient has cancer) and a false positive rate of 9% (ie. it is correct 91% of the time when the patient doesn’t have cancer). A patient takes this test and gets a positive result. What are the odds they have breast cancer? https://www.youtube.com/watch?v=lG4VkPoG3ko
Gigerenzer, Gerd, et al. "Helping doctors and patients make sense of health statistics." Psychological science in the public interest 8.2 (2007): 53-96.
Proof Building Blocks
6
AssertionsX ≠ Y
Z is an integerP is a prime
ConnectivesOr Introduction
Associativity of AdditionMathematical Induction
Proof Building Blocks
7
AssertionsX ≠ Y
Z is an integerP is a prime
(Axioms explicitly stated by the question)
ConnectivesOr Introduction
Associativity of AdditionMathematical Induction
(Axioms implied by the question)
Patterns of Proofs
9
Proof by Construction
Proof by Contradiction
Proof by Contrapositive
Proof by Induction
Proof by Contradiction
Assertion: There are an infinite number of primes.Proof: ● Assume a finite number instead.● Collect them all in an ordered list: { p1, p2, … pn }● Calculate 1 + ∏ pk● Not divisible by any prime, must be prime.● Bigger than pn .● Contradiction!
Proof by Contradiction?
Assertion: A = BProof: ● Assume A ≠ B● (proof by construction)● Contradiction!
Proof By Contrapositive via Proof By Contradiction
A ⇒ BAssume ¬B
Assume AB (premise 1)
⚔ (assumption 1)
¬A (non-contradiction)
¬B ⇒ ¬A (premise 1 and contradiction)
Proof by Contrapositive
Modus tollens is in the form, If P, then Q; not Q; therefore not P. Thus, whenever the consequent (Q) gets denied in the second premise, it leads to denying the antecedent (P) in the conclusion. Such operation ought to prevail with probabilistic premises, as well (e.g., Oaksford and Chater, 2001, 2009; Evans et al., 2015), whereby a probable consequent (Qp) may be denied without its probability warranting transposition onto a non-probabilistic antecedent (P). For example, if all red cars (P) have a 95% chance of getting stolen (Q ≥ 0.95) and we learn of a Lamborghini with a lesser probability of so disappearing (not Q ≥ 0.95), it is logical to conclude that the Lamborghini is not red (not P).
Perezgonzalez JD. Commentary: The Need for Bayesian Hypothesis Testing in Psychological Science. Front Psychol. 2017;8:1434. Published 2017 Aug 23. doi:10.3389/fpsyg.2017.01434 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5572393/
Proof by Induction, Inductive Case 1
Least Residuesqn < b
Proven b = (q-1)n + a
b - n = qn - n + ab - n = (q-1)n + a
b ≡ b - n ≡ a