DM_lecture1.5

Hui Gao
Discrete Mathematics
15
Counter Examples
1. 2. 3.
4.
5.
(x)(y)G(x, y) , where G(x, y)：y>x (y)G(y, y) US,(1) Wrong! (y)G(z, y) US,(1) (y)(y)G(y, y) UG,(3) Wrong! (z)(y)G(z, y) UG,(3)
Hui Gao
Discrete Mathematics
5
Modus Ponens

pq p q
p T T F F q p→q T T F F T T F T
Rule of modus ponens (a.k.a. law of detachment)
p T T F F q T F T F
Example: If I dance all night, then I get tired. I danced all night. Therefore I got tired.
§1.5 Rules of Inference


In mathematics, a proof is: a correct (well-reasoned, logically valid) and complete (clear, detailed) argument that rigorously & undeniably establishes the truth of a mathematical statement. Why must the proof be correct & complete? Correctness: If the system (claims to) prove something is true, it really is true.
Hui Gao Discrete Mathematics
13
More Examples
Show that S R can be derived from (P Q), (P R) and (Q S) Step Inference Rule (1) P Q P (2) P Q T, (1) (3) Q S P (4) P S hypo. Syll. on (2), (3) (5) S P T, (4) (6) P R P (7) S R T, (5), (6) (8) S R T, (7)

Hui Gao Discrete Mathematics
11
Proof Example cont.
Step 1. sunny cold 2. sunny 3. swim ↔ sunny 4. swim 5. swimcanoe 6. canoe 7. canoeearly 8. early Proved by Premise #1. Simplification of 1. Premise #2. IFF Truth table on 2,3. Premise #3. Modus ponens on 4,5. Premise #4. Modus ponens on 6,7.

Hui Gao
Discrete Mathematics
2
More Proof Terminology

Lemma - A minor theorem used as a steppingstone to proving a major theorem. Corollary - A minor theorem proved as an easy consequence of a major theorem. Conjecture - A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.) Theory – The set of all theorems that can be proven from a given set of axioms.
Hui Gao
Discrete Mathematics
12
that (RS) can be derived from (P (Q S)), (R P), and Q Step Inference Rule (1) R P P (2) R P (assumed premise) (3) P disjunctive syllogism on (1), (2) (4) P(Q S) P (5) Q S Modus ponens on (3), (4) (6) Q P (7) S Modus ponens on (5), (6) (8) R S conditional premise of (2), (7)
Hui Gao
Discrete Mathematics
14
Inference Rules for Quantifiers
x P(x) P(o) P(g) x P(x) x P(x) P(c) P(o) x P(x)

Universal instantiation (US) (substitute any object o) (for g a general element of u.d.) Universal generalization (UG) Existential instantiation (ES) (substitute a new constant c) (substitute any extant object o) Existential generalization (EG)
Hui Gao
Discrete Mathematics
10
Proof Example cont.
Let us adopt the following abbreviations: sunny = “It is sunny”; cold = “It is cold”; swim = “We will swim”; canoe = “We will canoe”; early = “We will be home early”. Then, the premises can be written as: (1) sunny cold (2) swim ↔ sunny (3) swim canoe (4) canoe early
Hui Gao
Discrete Mathematics
3
Inference Rules - General Form

An Inference Rule is

A pattern establishing that if a set of antecedent statements (hypotheses) are all true, then we can validly deduce that a certain related consequent statement (conclusion) is true.
Hui Gao
Discrete Mathematics
6
Modus Tollens

pq q p
Rule of modus tollens Rule of hypothetical syllogism Rule of disjunctive syllogism
Discrete Mathematics
7
pq qr pr pq p q

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Constructive dilemma

(P Q) (R S) PR QS (P Q) ( P S) PP QS
Constructive dilemma

Resolution
Hui Gao
Discrete Mathematics

Completeness: If something really is true, the
system is capable of proving it.

Proving a theorem allows us to rely upon on its correctness even in the most critical scenarios.
Hui Gao Discrete Mathematics
4
Some Inference Rules
p pq pq p p q pq

Rule of Addition
Rule of Simplification Rule of Conjunction
Hui Gao
Discrete Mathematics

Hui Gao Discrete Mathematics
9
Formal Proof Example

Suppose we have the following premises: “It is not sunny and it is cold.” “We will swim if only if it is sunny.” “If we do not swim, then we will canoe.” “If we canoe, then we will be home early.” Given these premises, prove the theorem “We will be home early” using inference rules.
Hui Gao Discrete Mathematics
1
Proof Terminology