Chapter 2 Fundamentals of Logic Discrete and Combinatorial Mathematics R. P. Grimaldi, 5th edition, 2004 Chapter 2 Fundamentals of Logic

Logic Logic = the study of correct reasoning Use of logic In mathematics: to prove theorems In computer science: to prove that programs do what they are supposed to do

Propositions (命題) A proposition or statement (陳述) is a declarative sentence that may be assigned a ‘true’ or ‘false’ value, but not both. This value is the truth value of the proposition. Propositions: “1+2=3”, “Peter is a programmer”, “It is snowing”. Not Propositions: “Is 1+2=3?”, “What a beautiful evening!”, “The number x is an integer”. Also Propositions: “There exists no ghost”.

True or False, That’s All The proposition “There exists no ghost” is on equal footing with the “1+2=3” proposition. The only thing that matters is the fact that a proposition is ‘True’ or ‘False’. Because of this, we will often label our propositions simply p,q, etc. Sometimes we use 0 for False and 1 for True. Things become interesting if we combine propositions…

Connectives (連接詞) If p and q are propositions, new compound propositions (複合命題) can be formed by using connectives. Most common connectives: Conjunction AND Symbol ^ (且) Disjunction OR Symbol v (或) Exclusive disjunction OR Symbol v (互斥或) Negation NOT Symbol  (非) Implication Symbol  (則，蘊含) Biconditional Symbol  (若且唯若) The truth values of compound propositions can be described by truth tables (真值表）.

Truth table of conjunction p ^ q is true only when both p and q are true. Example: p = "Tigers are animals", q = "Lions are plants" p ^ q = "Tigers are animals and Lions are plants" p q p ^ q 1

Truth table of disjunction The truth table of disjunction: p  q is false only when both p and q are false Example: p = "John is a programmer", q = "Mary is a lawyer" p v q = "John is a programmer or Mary is a lawyer" p q p v q 1

Exclusive disjunction “Either p or q” (but not both), in symbols p  q p  q is true only when p is true and q is false, or p is false and q is true. Example: p = "John is programmer, q = "Mary is a lawyer“ p v q = "Either John is a programmer or Mary is a lawyer but not both." p q p v q 1

Negation Negation of p: in symbols p p is false when p is true,  p is true when p is false. Example: p = "John is a programmer“  p = "It is not true that John is a programmer" p p 1

More compound propositions Let p, q, r be primitive propositions (簡單命題). We can form other compound propositions, such as (pq)^r p(q^r) ( p)( q) (pq)^( r) and many others…

Example: truth table of (pq)^r 1

Implication A conditional proposition (條件命題) is of the form “p implies q”, denoted by p  q, where p is the hypothesis (前提) and q is the conclusion (結論). Also “If p then q”.“p is sufficient for q”, “p is a sufficient condition (充分條件) for q”, “q is necessary for p”, “q is a necessary condition (必要條件) for p”, “p only if q”. Example: p = " John is a programmer " q = " Mary is a lawyer " p  q = “If John is a programmer then Mary is a lawyer"

Truth table of p  q p q p  q 1 p  q is true when both p and q are true or when p is false

Biconditional The biconditional proposition (雙向命題) is of the form “p if and only if q” or “ p iff q ” (若且唯若), denoted by p  q. p  q is true when both p  q and q  p are true. Example: p = " John is a programmer " q = " Mary is a lawyer " p  q = “John is a programmer iff Mary is a lawyer" p q p  q 1

Logical equivalence (邏輯等價) Two propositions are said to be logically equivalent () if their truth tables are identical. Example:  p  q  p  q p q  p  q p  q 1

Biconditional vs. Equivalence Don’t confuse the equivalence  with the biconditional  (only the biconditional has a truth table). For example: p  p is a proposition, a statement within logic, p  p is mathematically correct, … about logic. p  p is a False, p  p is incorrect Hence pp  (pp), and so on.

Converse The converse of p  q is q  p These two propositions are not logically equivalent p q p  q q  p 1

They are logically equivalent. Contrapositive (對換句) The contrapositive of the proposition p  q is  q   p. They are logically equivalent. p q p  q  q   p 1

Tautology (恆真命題) A proposition is a tautology (T0) if its truth table contains only true values for every case. p q p  p v q 1

Contradiction (矛盾命題) A proposition is a contradiction (F0) if its truth table contains only false values for every case. p p ^ ( p) 1

Example Pay attention to the phrase “logically”: “2 = 3–1” is not a tautology, but “2=1 or 21” is; “1+1=3” is not a contradiction, but “1=1 and 11” is. As with equivalence: look at the truth tables.

Proving Things in Logic The standard approach is to use truth tables. If we deal with n simple propositions p1,…,pn, our truth table will have size at least 2n. This becomes a substantial disadvantage if n is big. Sometimes there is a much more efficient way to prove equivalences. First, look at some very simple equivalences…

The Laws of Logic Double negation law: p  p       (雙否定定律) De Morgan’s laws: (pq)  pq, (De Morgan 定律) (pq)  pq Commutative laws: pq  qp and (交換律) pq  qp Associative laws: p(qr)  (pq)r, (結合律) p(qr)  (pq)r Distributive laws: p(qr)  (pq)(pr), (分配律)  p(qr)  (pq)(pr)

The Laws of Logic (Cont.) Idempotent laws: pp  p, (冪等定律) pp  p Identity laws: pFalse  p, (恆等定律) pTrue  p Inverse laws: pp  True, (逆定律) pp  False Domination laws: pTrue  True, (支配律) pFalse  False Absorption laws: p(pq)  p, (吸收律) p(pq)  p

Proving Equivalences Prove (p  q )  (p  q)  p.  (p  q )  (p  q) [DeMorgan’s Law]  (p  q )  (p  q) [Double Negation]  p  (q  q) [Distributive Law]  p  F0 [Inverse Law]  p [Identity Law] #

Simplify Statements Simplify “(p  q  r)  (p  t  q)  (p  t  r)”. (p  q  r)  (p  t  q)  (p  t  r) p  [(q  r)  (t  q)  (t  r)] [Distributive Law] p  [(q  r)  (t  r)  (t  q )] [Commutative Law] p  [((q  t)  r)  (t  q )] [Distributive Law] p  [((q  t)  r)  (t  q )] [Double Negation] p  [((q  t)  r)  (t  q )] [DeMorgan’s Law] p  [(t  q )  ((q  t)  r)] [Commutative Law] p  [((t  q )  (q  t))  ((t  q )  r)] [Distributive Law] p  [F0  ((t  q )  r)] [s  s  F0  s] p  [(t  q )  r] [F0 is the identity for ]

Simplify Statements (Cont.) p  [(t  q )  r] p  [(t  q )  r] [DeMorgan’s Law] p  [(t  q )  r] [Double Negation] p  [r  (t  q )] [Commutative Law] Hence (p  q  r)  (p  t  q)  (p  t  r)  p  [r  (t  q )]. #

Valid arguments (有效論證) Deductive reasoning（演繹推導）: the process of reaching a conclusion q from a sequence of propositions p1, p2, …, pn. The propositions p1, p2, …, pn are called premises or hypothesis. The proposition q that is logically obtained through the process is called the conclusion.

Rules of inference 1. Modus ponens (肯定前件式) p  q p Therefore, q Example: Insects have six legs. Beetles are insects. Therefore beetles have six legs.

Rules of inference (Cont.) 2. Modus tollens (否定後件式) p  q ~q Therefore, ~p Example: Insects have six legs. Spiders have eight legs. Therefore spiders are not insects.

Rules of inference (Cont.) 3. Rule of disjunctive amplification p Therefore, p  q 4. Rule of conjunctive simplification p ^ q Therefore, p 5. Rule of conjunction p q Therefore, p ^ q

Rules of inference (Cont.) 6. Law of the syllogism (三段論法) p  q q  r Therefore, p  r Example: 72 is divisible by 6. 6 is divisible by 3. Therefore 72 is divisible by 3.

Rules of inference (Cont.) 7. Rule of disjunctive syllogism (析取三段論法) p  q p Therefore, q Example: John is studying or sleeping. John is not studying. Therefore John is sleeping.

Rules of inference (Cont.) 8. Rule of contradiction (矛盾證法) p  F0 Therefore, p If we want to establish the validity of the argument (p1  p2  …  pn)  q, we can establish the validity of the logically equivalent argument (p1  p2  …  pn  q )  F0.

Example Demonstrate the validity of the argument ((p  r)  (p  q)  (q  s))  (r  s) (1) p  r (2) r  p (3) p  q (4) r  q (Law of the Syllogism) (5) q  s (6) Therefore r  s (Law of the Syllogism)

Example Demonstrate the validity of the argument (((p  q)  (r  s))  (r  t)  (t))  p (1) r  t (2) t (3) r (Rule of disjunctive syllogism) (4) r  s (Rule of disjunctive amplification) (5) (r  s) (De Morgan’s law) (6) (p  q)  (r  s) (7)  (p  q) (Rule of disjunctive syllogism) (8) p  q (De Morgan’s law) (9) Therefore p (Rule of conjunctive simplification)

Quantifiers (量詞) A propositional function (命題函數) or open statement P(x) is a statement involving a variable x. For example: P(x): 2x is an even integer, where x is an element of a set D.

Domain of a Propositional Function In the propositional function P(x): “2x is an even integer”, the domain or universe D of P(x) must be defined, for instance D = {integers}.D is the set where the x's come from.

For every and for some Most statements in mathematics and computer science use terms such as for every and for some. For example: For every triangle T, the sum of the angles of T is 180 degrees. For every integer n, n is less than p, for some prime number p.

Universal quantifier One can write P(x) for every x in a domain D In symbols: x P(x) “” is called the universal quantifier （通用量詞）. The statement x P(x) is True if P(x) is true for every x  D False if P(x) is not true for some x  D Example: Let P(n) be the propositional function n2 + 2n is an odd integer n  D = {all integers} P(n) is True only when n is an odd integer, False if n is an even integer.

Existential quantifier For some x  D, P(x) is true if there exists an element x in the domain D for which P(x) is true. In symbols: x, P(x) “” is called the existential quantifier （存在量詞）.

Counterexample The universal statement x P(x) is false if x  D such that P(x) is false. The value x that makes P(x) false is called a counterexample to the statement x P(x). Example: P(x) = "every x is a prime number", for every integer x. But if x = 4 (an integer) this x is not a primer number. Then 4 is a counterexample to P(x) being true.

Generalized De Morgan’s Laws If P(x) is a propositional function, then each pair of propositions in a) and b) below have the same truth values: a) ~(x P(x)) and x: ~P(x) "It is not true that for every x, P(x) holds" is equivalent to "There exists an x for which P(x) is not true“ b) ~(x P(x)) and x: ~P(x) "It is not true that there exists an x for which P(x) is true" is equivalent to "For all x, P(x) is not true"

Rules of inference for quantified statements 1. Universal instantiation  xD, P(x) d  D Therefore P(d) 2. Universal generalization P(d) for any d  D Therefore xD, P(x) 3. Existential instantiation  x  D, P(x) Therefore P(d) for some d D 4. Existential generalization P(d) for some d D Therefore  xD, P(x)

Equivalences and Implications for quantified statements For a prescribed universe and any open statements P(x) and q(x) in the variable x: x [p(x)  q(x)]  [x p(x)  x q(x)] x [p(x)  q(x)]  [x p(x)  x q(x)] x [p(x)  q(x)]  [x p(x)  x q(x)] [x p(x)  x q(x)]  x [p(x)  q(x)]

Demonstrate Universally Quantified Statement In order to prove the universally quantified statement x P(x) is true It is not enough to show P(x) true for some x  D You must show P(x) is true for every x  D In order to prove the universally quantified statement x P(x) is false It is enough to exhibit some x  D for which P(x) is false This x is called the counterexample to the statement x P(x) is true

Axioms (公設) An axiom is a proposition accepted as true without proof within the mathematical system. There are many examples of axioms in mathematics: Example: In Euclidean geometry the following are axioms Given two distinct points, there is exactly one line that contains them. Given a line and a point not on the line, there is exactly one line through the point which is parallel to the line.

Theorems (定理） A theorem is a proposition of the form p  q which must be shown to be true by a sequence of logical steps that assume that p is true, and use definitions, axioms and previously proven theorems.

Lemmas and corollaries A lemma (輔助定理） is a small theorem which is used to prove a bigger theorem. A corollary (引理） is a theorem that can be proven to be a logical consequence of another theorem. Example from Euclidean geometry: "If the three sides of a triangle have equal length, then its angles also have equal measure."

Types of proof A proof is a logical argument that consists of a series of steps using propositions in such a way that the truth of the theorem is established. Direct proof Indirect proof

Direct proof Direct proof: p  q A direct method of attack that assumes the truth of proposition p, axioms and proven theorems so that the truth of proposition q is obtained. Example: For all real numbers d, d1, d2, and x, if d=min{d1,d2} and xd, then xd1 and xd2. Proof. From the definition of min, it follows that dd1 and dd2. From xd and dd1, we may derive xd1 by the transitive property of . From xd and dd2, we may derive xd2 by the same property. Therefore, xd1 and xd2. #

Indirect proof The method of proof by contradiction of a theorem p  q consists of the following steps: 1. Assume p is true and q is false 2. Show that ~p is also true. 3. Then we have that p ^ (~p) is true. 4. But this is impossible, since the statement p ^ (~p) is always false. There is a contradiction! 5. So, q cannot be false and therefore it is true. Or show that the contrapositive (~q)(~p) is true. Since (~q)  (~p) is logically equivalent to p  q, then the theorem is proved.

Example of Indirect proof For all real numbers x and y, if x+y2, then either x1 or y1. Proof. Assume that x + y  2, x < 1 and y < 1. Then it follows that x + y < 1 + 1 = 2. It contradicts to the assumption x + y  2. Thus, we conclude that x1 or y1. #

Brainstorm 騎士在旅途中遇見四個人，來自兩個不同家族F1 和 F2。在這四人的談話中，如果是關於與自己同家族所說的話，則是真話；否則是謊話。四人A,B,C,D的對話如下: A: B屬於F1家族。 B: C屬於F1家族。 C: D屬於F2家族。 D: A屬於F2家族。 問題： 誰屬於F1家族？誰屬於F2家族？