CSS342: Proofs 1

CSS342: Proofs

Professor: Munehiro Fukuda

CSS342: Proofs 2

Terminologies• Axioms: are assumed true.

– Ex: Given two distinct points, there is exactly one line that contains them.

• Undefined terms: implicitly defined (and used) by the axioms.– Ex: Points, lines

• Definitions: used to create new concepts.– Ex: Two lines are parallel if they never cross each other.

• Theorem: a proposition that has been proved to be true.– If two sides of a triangle are equal, the angles opposite them are equal.

• Corollary: a theorem that follows quickly from another theorem.– Ex: If a triangle is equilateral, then it is equiangular.

• Lemma: a theorem not interesting but useful in proving another theorem. – Ex: A positive integer – 1 ≥ 0

CSS342: Proofs 3

Axioms, Definitions, and Undefined TermsExamples

• Euclidean geometry:– Axiom 1: Given two distinct points, there is exactly one line that contains them.– Axiom 2: If three points are not collinear, then there is exactly one plane that

contains them.– Definition 1: Two angles are supplementary if the sum of their measures is 180.– Definition 2: Two lines are parallel if they never cross each other.– Undefined terms: points, lines, planes, and angles

• Real numbers:– Axiom 1: The commutative law stands up right for +-*/ operations.– Axiom 2: If x and y are in a subset P, -x and –y are not in P, and x+y and xy are

in P.– Definition 1: P is called positive real numbers.– Definition 2: Given a nonnegative real number, x and a positive integer, n, x1/n is

y satisfying yn = x– Undefined terms: numbers and 0

CSS342: Proofs 4

Theorem and CorollaryExamples

• Theorem:– If two sides of a triangle are equal, then the angles opposite

them are equal.– Proof.

• Corollary:– If a triangle is equilateral, then it is equiangular.

Draw a line from the top vertex down to the

bottom side so that the side is divided into

two halves equally.

Then, we can have two sub triangles, both

having all equal sides.

Thus, they are congruent.

Therefore, the angles opposite them are equal.

CSS342: Proofs 5

LemmaExample

• If n is a positive integer number, the either n – 1 is a positive integer or n – 1 = 0.

• Proof:– The minimal positive integer is 1. Thus, n – 1 cannot be

smaller than 0. Therefore, the lemma is true.– This lemma is not interesting in its own right, but can be

used to prove other results.

CSS342: Proofs 6

Direct Proof

• When proving a universally quantified statement: x1, …, xn, P(x1, …, xn)→q(x1, …, xn)

• If P(x1, …, xn) is false, this statement is always true.

• Thus, focus on only the case when P(x1, …, xn) is true.

• Using P(x1, …, xn) for a proof is called

A direct proof

A

CSS342: Proofs 7

Direct ProofExample

• Universally quantified statement:– If d = min{d1, d2} and x ≤ d, x ≤ d1 and x ≤ d2

• Proof:Assume that d = min {d1, d2} and x ≤ d.

From the definition of min,

d ≤ d1 and d ≤ d2

From x ≤ d and d ≤ d1 , x ≤ d1

From x ≤ d and d ≤ d2 , x ≤ d2

Thus, the statement is true.

CSS342: Proofs 8

Proof by contradiction• Assume that the hypothesis p is true but the conclusion q

is false.• Use p, !q and r (= other axioms, definitions, theorems).• Derive r && !r = false. In other words, p && !q→ r && !r

p q (!q) r p → q p&&!q r&&!r p && !q → r && !r

T T (F) T T F F T

T T (F) F T F F T

T F (T) T F T F F

T F (T) F F T F F

F T (F) T T F F T

F T (F) F T F F T

F F (T) T T F F T

F F (T) F T F F T

CSS342: Proofs 9

Proof by ContradictionExample 1

• If xy = 0, then either x = 0 or y = 0• Proof.

– p: xy = 0– q: x = 0 || y = 0– !q: !(x=0 || y=0) ≡ x ≠ 0 && y ≠ 0– r: if ab = ac and a ≠ 0, b = c (Let’s assume it has been

proved)– p && !q: xy = x * 0 = 0 and x ≠ 0 – From r: y must be 0– This contradicts !q, which thus means r is wrong.– This derives r && !r– Thus, this statement must be true.

CSS342: Proofs 10

Proof by ContradictionExample 2

• For all real numbers x and y, if x + y ≥ 2, then either x ≥ 1 or y ≥ 1.

• Proof. – p: x + y ≥ 2– q: x ≥ 1 || y ≥ 1– !q: !(x ≥ 1 || y ≥ 1) ≡ !(x ≥ 1) && !(y ≥ 1) ≡ x < 1 && y < 1 ≡ x +

y < 2 ≡ !p– This derived p && !p– Thus, the statement must be true.– Therefore, the statement must be true.

• Instead of r && !r, we derived p && !p, (i.e., r = p)– Special case: proof by contrapositive

CSS342: Proofs 11

Deductive Reasoning

• Drawing a conclusion from a sequence of propositions.• Hypothesis:

P1: The bug is either in module 17 or in 81.

P2: The bug is a numerical error.

P3: Module 81 has no numerical error.

• Conclusion: ∴ Q: The bug is in module 17

CSS342: Proofs 12

p→qp ∴ q

• P: 1 * 2 = 2• Q: I ate candy.• P is true• Thus Q is true (= I ate candy)

p q p → q p → q && p qT T T T TT F F F FF T T F TF F T F F

Note: p →q, p /∴ q does not mean p →q && p ≡ ∴ q

In fact, truth values do not match perfectly.

It means that if p →q && p, then q = true.

CSS342: Proofs 13

p→qq ∴ p

• P: 1 + 2 = 2• Q: I ate my hat.• Q is true (= I ate my hat.)• Then, is 1 + 2 = 2 true?

p q p → q p → q && q pT T T T TT F F F TF T T T FF F T F F

When p → q && q = true, p can be true or false.

Thus, this deductive argument is wrong.

CSS342: Proofs 14

Rules of inference for propositions

Rule of Inference

Name Rule of Inference

Name

p → q

p

∴q

Modus ponens p

q

∴p && q

Conjunction

p → q

!q

∴!P

Modus tollens p → q

q → r

∴p → r

Hypothetical syllogism

p

∴p || q

Addition p || q

!p

∴q

Disjunctive syllogism

p && q

∴p

Simplification

CSS342: Proofs 15

Inference Example 1

• If you pass CSS342, then you can take CSS343. (p → q)• If you can take CSS343, then you’ll learn binary trees. (q → r)• If you can take CSS343, then you’ll learn inheritance. (q → s)• You passed CSS342. (p)

• Applying the hypothetical syllogism:

• Thus, r && s• You’ll learn both binary trees and inheritance

p → q

q → r

∴p → r

p → q

q → s

∴p → s

CSS342: Proofs 16

Rules of Inference for Quantified Statements

Rules of Inference Name

∀x D P(x)∈∴P(d) if d D∈

Universal instantiation

P(d) for any d D∈∴∀x P(x)

Universal generalization

∃x D P(x)∈∴P(d) for some d D∈

Existential instantiation

P(d) for some d D∈ ∴ ∃ x P(x)

Existential generalization

CSS342: Proofs 17

Inference Example 2

• Given two statements:– Everyone loves either Microsoft or Apple.– Lynn does not love Microsoft.

• P(x): x loves Microsoft.• Q(x): x loves Apple.• From universal generalization,

– ∀x P(x) || Q(x): Everyone loves either MicroSoft or Apple.

• !P(Lynn): Lynn does not love Microsoft.• From disjunctive syllogism: p || q, !p / q∴

– Q(Lynn) is true.

• Q(Lynn): Lynn loves Apple.

CSS342: Proofs 18

Final ReviewWhy Logic is So Important in CS?

• Inference is quite often used in knowledge database.• Knowledge database is a core of expert system.• Thus, inference is a core of expert system. • Example in CS: Prolog

likes(mary, food).

likes(mary, wine).

likes(john, wine).

likes(john, mary).

?- likes(mary, X), likes(john, X).

X=wine