Discrete Mathematics Logic

Propositions A proposition is a statement or sentence that can be determined to be either true or false (but no both). Examples: The only positive integers that divide 7 are 1 and 7 itself. Buy two tickets for Friday concert. Earth is the only planet in the universe that contains life.

Example Use variable to represent propositions P: 1+1=3 P: It is raining outside P: Today is Tuesday

Connectives If p and q are propositions, new compound propositions can be formed by using connectives Most common connectives: Conjunction (and) ^ Disjunction (or)  Negation (not) ~ Exclusive-OR v Condition (if … then)  Bi-Condition 

Example P: It is raining Q: It is cold Form a new compound statement by combining these two statements P  Q : It is raining and it is cold P  Q : It is raining or it is cold

Truth table of conjunction The truth values of compound propositions can be described by truth tables. Truth table of conjunction P Q is true only when both P and Q are true. P Q P  Q T F

Example Let P = “A decade is 10 years” Let Q = “A millennium is 100 years” P  Q = “A decade is 10 years” and “A millennium is 100 years” If P is true and Q is false then conjunction is false

Truth table of disjunction The truth table of disjunction is 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  Q T F

Negation Negation of P: in symbols ~P or ⌐P ~P is false when P is true, ~P is true when P is false Example, P : "John is a programmer" ~P = "John is not a programmer" P ~P T F

E.g P: Paris is the capital of England ~P: Paris is not capital of England

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" P Q P v Q T F

More compound statements Let p, q, r be simple statements We can form other compound statements, such as (pq)^r p(q^r) (~p)(~q) (pq)^(~r) and many others…

Example: truth table of (P Q)R

Conditional propositions A conditional proposition is of the form “If P then Q” In symbols: P  Q Example: P = " A bottle contains acid" Q = “A bottle has a label” P  Q = “If a bottle contains acid then it has a label "

P  Q is true when both p and q are true Truth table of P  Q P Q P  Q T F P  Q is true when both p and q are true or when P is false

Example If the mathematics department gets an additional $40,000 then it will hire one new faculty member. Let P: The Mathematics Department gets an additional $40,000 and Q: The mathematics Department will hire one new faculty member.

Hypothesis and conclusion In a conditional proposition P  Q, P is called the hypothesis Q is called the conclusion

Example For all real number x if x > 0 then x2 > 0 For example x=3 , 3 > 0 then 32 > 0 both are true. x=-2 , -2 > 0 is false but -22 > 0

Logical equivalence Two propositions are said to be logically equivalent if their truth tables are identical. Example: ~P  Q is logically equivalent to P  Q P Q ~P  Q P  Q T F

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

They are logically equivalent. Contrapositive The contrapositive of the proposition p  q is ~q  ~p. They are logically equivalent. p q p  q ~q  ~p T F

p  q is logically equivalent to (p  q)^(q  p) Bi-Conditional The double implication “p if and only if q” is defined in symbols as p  q p  q is logically equivalent to (p  q)^(q  p) p q p  q (p  q) ^ (q  p) T F

Tautology A proposition is a tautology if its truth table contains only true values for every case Example: p  p v q p q p  p v q T F

Contradiction A proposition is a tautology if its truth table contains only false values for every case Example: p ^ ~p p p ^ (~p) T F

De Morgan’s laws for logic The following pairs of propositions are logically equivalent: ~ (p  q) and (~p)^(~q) ~ (p ^ q) and (~p)  (~q)

Quantifiers A propositional function P(x) is a statement involving a variable x For example: P(x): 2x is an even integer x is an element of a set D For example, x is an element of the set of integers D is called the domain of P(x)

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

Inference Or Deduction An argument is a sequence of propositions written as below : p1, p2 , p3 , p4 , p5 ,,…, , pn, therefore q. (if p1v p2v,.., pn then q) or (if p1 ^p2 ^,.., pn then q) the propositions p1, .., pn, are called hypotheses (premises) and proposition q is called conclusion (consequent). The above argument is valid if the consequent can be proven from the premises. Validity can be shown using truth table or by using theorems / rules.

Rules of Inference Addition p , therefore p v q Simplification (read as if we know p is true, therefore we know p or q is true) Simplification p ^ q , therefore p Conjunction p , q , therefore p ^ q

Rules of Inference (contd..) Modus Ponens p  q , p, therefore q Modus Tollens ~ q, p  q , therefore ~p Hypothetical Syllogism p  q, q  r, therefore p  r Disjunctive Syllogism p v q , ~ p, therefore q Resolution p v q, ~ p v r , therefore q v r

Example State which rule of inference is the basis of the following argument : “ It is below freezing now. Therefore it is either below freezing or raining now.” - addition rule “It is below freezing and raining now. Therefore, it is below freezing now. “ - simplification rule “Rules : If it snows today, then we will go skiing” Fact : it snows today. Conclude: We will go skiing” - modus ponens

Problem Show that the hypotheses : “ It is not sunny this afternoon and it is colder than yesterday.” Will conclude : “We will be home by sunset “ Given the following fact : “We will go swimming only if it is sunny. If we do not go swimming, then we will take a canoe trip. If we take a canoe trip we will be home by sunset”