Lesson 4.1 Introduction to Logic PDF

LESSON 4.1 Introduction to  What is Logic is the study of method and principles used in distinguishing correct (good) from incorrect (bad) arguments. A proposition or (a statement) is a declarative sentence which is either true or false, but not both. The truth or falsity of a statemen...

LESSON 4.1 Introduction to  What is Logic is the study of method and principles used in distinguishing correct (good) from incorrect (bad) arguments. A proposition or (a statement) is a declarative sentence which is either true or false, but not both. The truth or falsity of a statement is called its truth value. The truth value of a proposition is true, denoted by T if it is a true statement, otherwise, the truth value is false, denoted by F. Propositional variables are used to represent propositions, usually denoted by small letters, such as p, q, r, s and t. Example: p: Everyone should study logic. p is the proposition “Everyone should study logic” Determine whether each of the following is a proposition or not, if a proposition, give its truth value. 1. p: Manila is the capital city of the Philippines. p is a true proposition 2. q: Find a number which divides your age. q is not a proposition 3. r: Zero is a rational number. r is a true proposition 4. s: Cats can fly. s is a false proposition Determine whether each of the following is a proposition or not, if a proposition, give its truth value. 5. t: Where are you going? t is not a proposition 6. : 6 is an even number is true proposition 7. : 9 is a prime number is a false proposition An open sentence contains one or more variables, that is, it is either true or false depending on the value of the placeholder. CONSIDER THE FOLLOWING OPEN SENTENCES: 1.She was the first Miss Philippines. 2.x is less than 10 3. 4. 5. He was the father of problem solving. A closed sentence, on the other hand, is a mathematical sentence that is known to be either true or false. 1. 9 is an odd number. 2. 4 + 4 = 8 3. 4. 5. The square root of 4 is 1. A compound proposition is a proposition formed from simple propositions using logical connectors or some combinations of logical connectors. Logical connectors involving propositions p and/or q may be expressed as: not p if p then q p and q p if and only if q p or q A proposition is simple if it cannot be broken down any further into other component propositions. Example: For each of the propositions, determine whether it is a simple or compound proposition. If it is compound proposition, identify the simple components. Simple or Compound Proposition? : If you study hard, then you will get good grades. h : You study hard. g : You get good grades.  : If h, then g. Simple or Compound Proposition? : Either logic is fun and interesting, or it is boring. f : Logic is fun i : Logic is interesting b : Logic is boring.  : f and i, or b QUANTIFIERS: Quantifiers are words, expressions, or phrases that point out the number of elements that a statement relates to. There are two types of quantifiers: universal and existential quantifier. UNIVERSAL QUANTIFIERS: The universal quantifier, denoted by refers to the phrase “for all’ or “for every” or “for each”. Let P(y) be a formula defined on a set D. Then the expression or y P(y) is read as “For each y in D, P(y) is a true statement. UNIVERSAL QUANTIFIERS: The universal quantifier asserts that the formula holds for any value of y (the value as being taken from some given universe or the set of objects of interest). Examples: Determine the truth value of the following statements. Define R to be the set of real numbers, N the set of natural numbers, and Z the set of integers. 1. The statement is true since the square of any real number is always nonnegative. Examples: Determine the truth value of the following statements. 2. The statement is false since 2 is a natural number and 3. The statement is true since the real numbers are commutative under addition. EXISTENTIAL QUANTIFIERS: The existential quantifier, denoted by refers to the phrase “there exists” or “for at least one” or “for some”. Let P(x) be a formula defined on a set D. Then the expression or is read as “There exists y in D such that P(y) is a true statement”. EXISTENTIAL QUANTIFIERS: The existential quantifier asserts that the formula holds for at least one value of y (the value as being taken from some given universe). Examples: Determine the truth value of the following statements. Define R to be the set of real numbers, N the set of natural numbers, and Z the set of integers. 1. The statement is true since if we choose then Examples: Determine the truth value of the following statements. 2. The statement is true since if we choose a natural number, then. 3. The statement is false since must be equal to to make the equation true, and is not an integer. BASIC L O G I CA L O P E RAT O R S : Given a proposition, its truth table shows all its possible truth values. A proposition p would have For proposition p and q, truth the truth table. value would be: p q p T T T T F F F T F F In general, a truth table involving n propositions has rows.  BASIC L O G I CA L O P E RAT O R S : Negations Let p be a proposition. The negation of p, denoted by is the proposition “It is not the case that p”, is read as “the negation of p” or “not p”. Definition: If p is true, then is false; and if p is false, then is true. p Truth Table for Negation T F F T BASIC L O G I CA L O P E RAT O R S : Examples: Write the negation of each of the following statements. 1. Manila City is in the Philippines. Negation: Manila is not in the Philippines. 2. Mary is a girl. Negation: Mary is a boy. 3. The product of two odd integers is odd. Negation: The product of two odd integers is even. BASIC L O G I CA L O P E RAT O R S : Examples: Write the negation of each of the following statements. 4. 3 times 7 = 20 Negation: 5. January has 31days Negation: January does not have 31 days. 6. Eleven is not a prime number. Negation: Eleven is a prime number. BASIC L O G I CA L O P E RAT O R S : Conjunctions A compound statement formed by connecting two propositions with the word “and’ is called a conjunction. In symbols, it is written as “ which is read as BASIC L O G I CA L O P E RAT O R S : Conjunctions Definition: If p and q are true, then is true; otherwise is false. Truth Table of Conjunction p q T T T T F F F T F F F F BASIC L O G I CA L O P E RAT O R S : Examples: Determine the truth value of each the following conjunctions. 1. Manny Pacquiao is a boxing champion and Rodrigo Duterte is the current Philippine President. Answer: Since the propositions “Manny Pacquiao is a boxing champion” and “Rodrigo Duterte is the current Philippine President” are both true, thus the conjunction is true. BASIC L O G I CA L O P E RAT O R S : 2. and 50 is divisible by 3. Answer: Since “ ” is a true proposition and “50 is divisible by 3” is false, the conjunction of the compound proposition is false. BASIC L O G I CA L O P E RAT O R S : 3. The earth is triangle and the moon is square. (False) 4. Square has four sides and snakes are mammal. (False) 5. and is an irrational number. (True) BASIC L O G I CA L O P E RAT O R S : Disjunctions A compound statement formed by connecting two statements with the word “or” is called a disjunction. Symbolically, “ which is read as BASIC L O G I CA L O P E RAT O R S : Disjunctions Definition: If p and q are false, then is false; otherwise is true. Truth Table of Disjunction p q T T T T F T F T T F F F BASIC L O G I CA L O P E RAT O R S : Examples: Determine the truth value of each the following disjunctions. 1. Manila City is in the Philippines or China is in the Philippines. Answer: Since the propositions “Manila City is in the Philippines” is true, and “China is in the Philippines” is false, hence the disjunction of the compound proposition is true. BASIC L O G I CA L O P E RAT O R S : 2. is a negative integer or is a positive integer. Answer: Since the propositions “ is a negative integer” and is a positive integer” are both false, thus the disjunction of the compound proposition is false. BASIC L O G I CA L O P E RAT O R S : 3. Chocolate is sweet or today is Sunday. (True) 4. Chicken is not a bird or the freezing point of water is 100. (False) 5. October has 31 days or 2028 is a leap year. (True)  BASIC L O G I CA L O P E RAT O R S : Conditional Statement A compound statement formed by connecting two statements with the words “if…,then” is called a conditional. Symbolically, “ which is read as or “p implies q”. The statement p is called the antecedent (or the hypothesis or premise) and statement q is the consequent (or the conclusion) of the conditional.  BASIC L O G I CA L O P E RAT O R S : Conditional Statement In this case, the resulting sentence is only false whenever the antecedent is true and the consequent is false, and is true otherwise. Truth Table of Conditional Statement p q T T T T F F F T T F F T Examples: Construct the conditional statement and write its truth value. 1.The antecedent, p, is “A square is a quadrilateral”. And the consequent, q, is “A square has four sides.” Answer: If a square is a quadrilateral, then a square has four sides. (True because p and q are true) Examples: Construct the conditional statement and write its truth value. 2. The antecedent, p, is “Vinegar is sweet ” and the consequent, q, is “Sugar is sour.” Answer: If vinegar is sweet, then sugar is sour. (True because p and q are false) Examples: Construct the conditional statement and write its truth value. 3.The antecedent, p, is “I am studying hard”. And the consequent, q, is “I will not get good grades.” Answer: If I am studying hard, then I will not get good grades. (False because p is true and q is false)  BASIC L O G I CA L O P E RAT O R S : Conditional Statement Each of the following phrases is equivalent to the conditional If p, then q. Not p unless q. q follows from p. p only if q. p implies q. q if p. q whenever p. Whenever p, q q is necessary for p. p is sufficient for q.  BASIC L O G I CA L O P E RAT O R S : Biconditional Statement A compound statement formed by connecting two statements with the words “if and only if ” is called a biconditional. Symbolically, “ which is read as. The biconditional statement is actually formed by the conjunction of the conditional statements and  BASIC L O G I CA L O P E RAT O R S : Biconditional Statement In this case, the resulting sentence is true whenever the antecedent p as well as the consequent q have the same truth values and false otherwise. p q T T T Truth Table of Biconditional Statement T F F F T F F F T Examples: Construct the biconditional statement and write its truth value. 1.The antecedent, p, is “17 is a prime number”. And the consequent, q, is “17 is not an integer” Answer: 17 is a prime number if and only if 17 is not an integer. (False because p is true and q is false) Examples: Construct the biconditional statement and write its truth value. 2. The antecedent, p, is “Fishes live in the moon”. And the consequent, q, is “Birds can fly”. Answer: Fishes live in the moon if and only if birds can fly. (False because p is false and q is true) Examples: Construct the biconditional statement and write its truth value. 3. The antecedent, p, is “I am breathing”. And the consequent, q, is “I am alive” Answer: I am breathing if and only if I am alive. (True because p and q are true) BASIC L O G I CA L O P E RAT O R S : Exclusive-or The exclusive-or of the proposition p and q is the compound proposition “p exclusive-or q”. Symbolically, p  BASIC L O G I CA L O P E RAT O R S : Exclusive-or Definition: If p and q are true or both false, then p p. p q T T F Truth Table of Exclusive-or Statement T F T F T T F F F Examples of exclusive-or 1. You passed or you failed.” 2. Lights are switched on or lights are switched off. 3. She is single or she has a love life. 4. It is daytime or nighttime. 5. The sun rises or the sun sets. Determine if the statement is true or false. 1. Coconut provides shelter and sun gives light. TRUE Determine if the statement is true or false. 2. Manny Pacquiao is a boxer if and only if December is a summer month in the Philippines. FALSE Determine if the statement is true or false. 3. Cigarette smoking causes cancer or Filipino is the universal language, and Asia is a continent. TRUE Determine if the statement is true or false. 4. If ice cream is sweet, then cake is bitter. FALSE Determine if the statement is true or false. 5. Success follows from being hardworking. TRUE Determine if the statement is true or false. T 1. 2016 is a leap year. T 2. If March is the third calendar month, then one foot has 12 inches. T 3. February has 29 days whenever 2020 is a leap year. F 4. The boiling point of water is or the freezing point is 100. T 5. 20 is divisible by 2 if and only if 20 is even.