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## " MATHEMATICAL LOGIC "

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**" MATHEMATICAL LOGIC "**• MAIN TOPIC : • Statement, Open Sentences, and Truth Values • Negation • Compound Statement • Equivalency, Tautology, Contradiction, and Contingency • Converse, Inverse, and Contraposition • Statement with Quantor • Making Conclusion • Direct and Inderct Proof**The Statements, Open Sentences, and Truth Values**• Statement and Open Sentences • Definition of proposition : • “ Proposition is a sentence, which can explain something true or false ” Definition of open sentence : “ Open sentence is the sentence, not proposition. We can declare the open sentence as a proposition by changing the variabel with certain value “**Example !**• Determine whether the following sentences are propositions! • a. Jakarta is the capital city of Indonesia. • b. The root of the equation 2x + 8 = 0 is 4 • c. Is a 2 a prime number? • d. 2x + 6 = 0 • Answer : • a. The sentence (a) declares something, that is Jakarta can be either the capital • city of Indonesia or not. Actuallly, Jakarta is the capital city of Indonesia. • Thus, that sentence is a true proposition. • b. By solving the equation 2x + 8 = 0 we will get x = -4 which is the root of • that equation. Thus, that sentence is a false proposition. • c. The sentence (c) doesn’t explain, but ask something. Thus, that sentence is • not a proposition, but a question. • d. The sentence (d) is not a proposition. If we subtitude x = -3, we will get a • true proposition, but if we subtitude x= 2, we will get false propotion. This • kind of sentence is known as open sentence.**2. Which sentences are including open sentences! Solve those**open sentences! a. 2x + 2 < 5 , x Єnatural number. b. sin 30° = 1 √ 2 2 Answer : a. 2x + 2 < 5 , x Єnatural number is an open sentence. If x = 1 , then 2 . 1 + 2 < 5 is a true proposition. If x = 2 , then 2 . 2 + 2 < 5 is a false proposition. Thus, x = 1 is a solutionof 2x + 2 < 5 whereas x = 2 or x greater than 2 is not a solution. So the solution set is {1}. b. sin 30 ° = 1 √2 is not open sentence, but rather a false proposition because 2 sin 30˚ = 1 . 2 Example !**Notation and the Truth Value of a Proposition**In mathematical logic, a proposition is usually written by the small letters p,q,r, … etc. For example, proposition “ 12 + 3 = 5 “ and proposition “ Human Is breathing with lungs “ can be written respectively as follows. p : 12 + 3 = 5 q : Human is breathing with lungs. From the defenition of proposition, we understand that a propotition can be only true or false. Thus, the truth value of a proposition is only true ( T) or false (F). The truth value of a proposition can be denoted as . For the example above, we have : (p) : F ( read : the truth value of p is false ) (q) : T ( read : the truth value of q is true )**Determine the truth value of the following**• propotition! • 1. p : The sum of angels in any triangles is 360˚. • 2. q : 27 can be divided by 5. • 3. r : √2 is a irational number. • 4. s : One hour is equal to 360 seconds. • Answer : • (p) = F • (q) = F • (r) = T • (s) = F Example !**Negation**The negation from the first propotition is called negation from the first propotition. The notation of the negation from the propotition p is ~p. If the propotition ~p is false and conversely if p is false, then ~p would be true. As seen beside, we can understand the proposition and its negation by using The truth table. TheTruth Table of Proposition and Its Negation : p~p TF FT**Example!!**• State the regation of the following propositions, and then determine their truth • values! • 1. 2 is a prime number. • 2. The symmetry line of y = x² - 4x – 5 is x =5 • 3. x² + 5x + 10 = 0 has two distinct roots. • Answer : • p : 2 is a prime number. • ~p : 2 is not a prime number. • Because (p) = T, then (~p) = F. • q : The symmetry line of y = x² - 4x – 5 is x =5 • ~q : Not true that symmetry line of y =x² - 4x – 5 is x =5 • Because (q) =F, then (~q) = T • r : x² +5x +10 = 0 has two distinct roots. • ~r : x²+5x+10 = 0 has not two distinct roots. • Because (r) = F, then (~r) = T.**Compound Statement**Definition : A compound statement is a sentence that contains at least two simple prepotitions. Compound statement is constructed from simple propositions using connectives such as AND; OR; IF … THEN …; and … IF AND ONLY IF … In mathematical logic, the four connectives above respectively are written as AND (٨); OR ( v ); IF … THEN … ( → ); and … IF AND ONLY IF … ( ↔ ).**“ The FormsofCompoundStatement“**1. Conjunction 2. Disjunction 3. Implication 4. Biimplication**Conjunction**Conjunction is a kind of compound statements,, which uses a connecticve “AND”. Conjunction of two simple statements p and q denoted as p ٨q and read as : p and q. Conjunction is a logical operation that results in a value of true if both of its operands are true, otherwise a value of false. Look at the truth of Conjunction beside. The Truth Table of Conjunction p٨q p q p ٨q T T T T F F F T F F F F**" CONTOH "**• Find the truth values of the following conjunctions! • The Java Island is greater than Irian Island and 2 is an even number • The quadratic equation x² - ( 3k + 2 )x + ( 2k² + 3k ) = 0 has two real numbers • and Semarang is the capital of Central Java. • Answer : • p : The Java Island is greater than Irian Island. • q : 2 is a even number. • Based on the truth table of conjunction, because (p) = F and (q) = T, then • (p٨q) = T. • p : The quadratic equation x² - ( 3k + 2 )x + ( 2k² + 3k ) = 0 has two real numbers • roots. • q : Semarang is the capital of Central Java. • Based on the truth table of conjunction, because (p) = T and (q) = T, then • (p٨q) = T.**Disjunction**Disjunction is a kind of compound statements, which uses a connective “BUT”. Disjunction of two simple statements p and q denoted as p v q and read as p or q. Disjunction is logical operation that produces a value of false if and only if both of its operand are false. Thus, the disjunction will result in true whenever one or more of its operands are true. Look at the truth table of disjunction beside. The Truth Table of Disjunction p v q p q p v q T T T T F T F T T F F F**C o n t o h . . .**• Find the truth values of the following disjunctions! • ²log 3 . ³log 8 = 3 or Yogyakarta is an educational city. • The minimum value of ƒ (x) = -2 sin x is -1 or the solution set of 3 2x-2 = 1 is{ 0 }. • 3 • Answer : • p : 2 log 3 . 3 log 8 = 3 • q : Yogyakarta is an educational city. • Based on the truth table of disjunction, because (p)= T and (q)= T; then • (p v q) = T. • 2. p : The minimum value of f (x) = -2 sin x is -1. • q : The solution set of 3 2x-2 = 1 is { 0 }. • 3 • Based on the truth table of disjunction, because (p)= F and (q)= F • ; then (p v q)= F.**Implication**Implication or conditional statement is a kind of compound statements, which uses A connective “IF … THEN … “. The implication of two simple statements p and q denoted as p → q and read as if p then q. In such an implication, p is called the antecedent, while q is called the consequent. Implication will produce a value of false only in the case, the first operand (antecedent ) is true and the second operand ( consequent ) is false. Look at the truth table of implication beside. The Truth Table of Implication p → q pq p→q T T T T F F F T T F F T**“ NOTE “**• The implication p → q can read as : • Jika p, then q • p only if q • q if p • p is sufficient condition for q • q is necessary condition for p**" Contoh "**• Find the truth values of the following implications! • If log 5 + log 15 = log 20, then log 15 – log 5 = log 3 • If 1 = 1 √5 + 1 √2, then cos 30 ˚ = 1 • √5 - √2 3 3 2 • Answer : • p : log 5 + log 15 = log 20 • q : log 15 – log 5 = log 3 • Based on the truth table of implication, because (p) = F and (q) = T, • thus (p→q) =T. • p : 1 = 1 √5 + 1 √2 • √5 - √2 3 3 • q : cos 30˚ = 1 • 2 • Because on the truth table of implication, because (p) = T and (q) = F, thus • (p→q) = F**Biimplication**Biimplication of two simple statements p and q denoted as p↔q read as p if and only if q. A biimplication will produce a value of true only in the case the first operand (antecedent ) and the second operand ( consequent ) have the same truth value, whether true or false. Look at the truth table of biimplication beside. The Truth Table of Biimplication p → q p q p↔q T T T T F F F T F F F T**CONTOH !!!!**• Find the truth values of the following biimplications! • 2 log 16 = 4 if and only if 24 = 16 • x2 – x – 2 = 0 has two real roots if and only if x2 – 2x + 6 = 0 has real root. • Answer : • p : 2 log 16 = 4 • q : 2 4 = 16 • Based on the truth table of biimplication, because (p) = T and (q) = T • thus (p↔q) = F. • p : x2 – x – 2 = 0has real root • q : x2 – 2x + 6 = 0 has real root • Based on the truth table of biimplication, because (p) = T and (q) = F • thus (p↔q) =F.**Equvalence, Tautology, Contradiction,**and Contingency Definition : Two compound statements A and B are logically equivalent if thet have The same truth, denoted by A B. The Truth Table of Equivalence p q ~p p→q ~p v q T T F T T T F F F F F T T T T F F T T T Truth Table