Lesson 3 (Symbolic Logic) Concept Notes.pdf

Aklan Catholic College Arch. G.M. Reyes St. 5600 Kalibo, Aklan, Philippines Tel Nos.: (036) 268-40...

Aklan Catholic College Arch. G.M. Reyes St. 5600 Kalibo, Aklan, Philippines Tel Nos.: (036) 268-4010 Website: http://www.acc.edu.ph E-mail Add: [email protected] 1 EVALUATING THE VALIDITY OF LOGICAL STATEMENTS AND SYMBOLIC LOGIC Lesson 3 I. Intended Learning Outcomes At the end of this lesson, the learner will have 1. identified basic concepts in symbolic logic; 2. translated logical statements to symbols and vice versa; 3. constructed a truth table; 4. analyzed conditional statements by writing its converse, inverse and, contrapositive, and; 5. evaluated the validity of a logical statement. II. General Instructions 1. Read the guide questions 2. Study the concept notes. Refer to the materials cited for further understanding. 3. Study supplementary materials for additional information III. Guide questions Use this guide questions to navigate through the keynotes and additional readings and media. Keep them in mind while studying. You can use a separate note to pick up answers from the materials as move along with them. 1. Based on your own learning experience, how would you relate mathematics to the idea of logic? 2. What does it mean to reason mathematically? 3. How do you use logical reasoning to prove that statements are true? IV. Concept Notes Now that we have tackled the symbols of mathematics, we will now proceed with logic statements, logical connectives and symbols, Truth Value and Truth Table. We use the textbook Mathematics in the Modern World by Mary Joy J. Rodriquez, Ivy Gay O. Salvador, Feljone G. Ragma, et. al. (2018) as our main reference for this lesson. Let us begin our discussion by watching the video “No, no, you’re not thinking; you’re just being logical” by Martin Westwell of TEDxTalks (2015). This video is stored in the material section of GMathMod Google Classroom. What is Logic? Logic is the study of the principles of correct reasoning. It allows us to determine the validity of arguments in and out of mathematics. It illustrates the importance of precision and conciseness of the language of mathematics. What is Mathematical Logic? Mathematical Logic is a branch of mathematics with close connections to computers. It includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. What are Logic Statements? Every language contains different types of sentences, such as statements, questions and commands. For instance, “Is the test today?” is a question. “Go get the newspaper” is a command. “This is a nice car” is an opinion. “Kalibo is the capital of Aklan province” is a statement of fact. The symbolic logic that Boole was instrumental in creating applies only to sentences that are statements as defined below. What is a Statement? A statement is a declarative sentence that is either true or false, but not both true and false. Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  2 Examples: a. Florida is a state in the United States. b. How are you? c. 𝑥𝑥 + 1 = 5. Solution: a. Florida is one of the 50 states in the United States, so this sentence is true and it is a statement. b. The sentence “How are you?” is a question; it is not a declarative sentence. Thus it is not a statement. c. 𝑥𝑥 + 1 = 5 is a statement. It is known as an open statement. It is true 𝑥𝑥 = 4, and it is false for any other values of x. For any given value of x, it is true and false but not both. Logic Connectives and Symbols Statement Connective Symbolic form Type of statement not p Not ~𝑝𝑝 Negation p and q And 𝑝𝑝 ˄ 𝑞𝑞 Conjunction p or q Or 𝑝𝑝 ˅ 𝑞𝑞 Disjunction If p. then q If … then 𝑝𝑝 → 𝑞𝑞 Conditional p if and only if q if and only if 𝑝𝑝 ↔ 𝑞𝑞 Biconditional What is Truth Value and Truth Table?  The truth value of a simple statement is either true (T) or false (F).  The truth value of a compound statement depends on the truth values of its simple statements and its connectives.  A truth table is a table that shows the truth value of a compound statement for all possible truth values of its simple statements. What is Negation? The negation of the statement “Today is Friday.” is the statement “Today is not Friday.” In symbolic logic, the tilde symbol ~is used to denote the negation of a statement. If a statement p is true, its negation ~𝒑𝒑is true. Example: a. Ellie Goulding is an opera singer. b. The dog does not need to be feed. Solution: a. Ellie Goulding is not an opera singer. b. The dog needs to be feed. The Compound Statements in Symbolic Logic Consider the following simple statements. p: Today is Friday. q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic logic. 1. Today is Friday and it is raining. 2. It is not raining and I am going to a movie. 3. I am going to the basketball game or I am going to a movie. Solutions: 1. 𝒑𝒑 ˄ 𝒒𝒒 2. ~𝒒𝒒 ˄ 𝒓𝒓 3. ~𝒔𝒔 ˅ 𝒓𝒓 Write the following symbolic logic statements in words. 1. 𝑞𝑞 → 𝑠𝑠 2. ~𝑝𝑝 ˅ 𝑠𝑠 3. ~𝑟𝑟 ↔ ~𝑞𝑞 Solutions: 1. If it is raining, then I am not going to the basketball game. 2. Today is not Friday or I am not going to the basketball game. 3. I am not going to a movie if and only if it is not raining. Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  3 What are the different Logical Connectives? Mathematical statements may be joined by logical connectives which are used to combine simple propositions to form compound statements. These connectives are conjunction, disjunction, implication, biconditional and negation. Let p and q be propositions. 1. Negation The negation of a true statement is a false statement, and the negation of a false statement is a true statement. The truth table for negation is given below. 𝒑𝒑 ~𝒑𝒑 T F F T Example 1. Write the negation of the following statements: Statement Negation 𝑝𝑝: The sun rises in the morning. ~𝑝𝑝: The sun does not rise in the morning. 𝑞𝑞: 2.54 is an integer. ~𝑞𝑞: 2.54 is not an integer. 𝑟𝑟: 4 + 4 = 8. ~𝑟𝑟: 4 + 4 ≠ 8. 𝑠𝑠: 3 is an even integer. ~𝑠𝑠: 3 is not an even integer. We must show some caution in negating statements involving all, none and some. These words are referred to as quantifiers. Form of Statement Form of Negation all are some are not none are some are some are none are some are not all are 2. Conjunction The conjunction of the propositions p and q is the compound statement “𝑝𝑝 and 𝑞𝑞” denoted as 𝒑𝒑 ˄ 𝒒𝒒 which is true only when both 𝑝𝑝 and 𝑞𝑞 are true, otherwise, it is false. The conjunction is usually expressed as and, however, it can also be expressed as but, however, or, nevertheless. Truth Table for Conjunction 𝒑𝒑 𝒒𝒒 𝒑𝒑 ˄ 𝒒𝒒 T T T T F F F T F F F F Example 2. Given the truth values of 𝑝𝑝 and 𝑞𝑞, state the truth value of the indicated statements. a. 𝑝𝑝 true, 𝑞𝑞 false; ~𝑝𝑝 ˄ 𝑞𝑞 𝒑𝒑 𝒒𝒒 ~𝒑𝒑 ~𝒑𝒑 ˄ 𝒒𝒒 T F F F 3. Disjunction The disjunction of the propositions p and q is the compound statement “p or q” denoted as 𝒑𝒑 ˅ 𝒒𝒒 which is false only when both p and q are false, otherwise, it is true. Truth Table for Disjunction 𝒑𝒑 𝒒𝒒 𝒑𝒑 𝐯𝐯 𝒒𝒒 T T T T F T F T T F F F Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  4 4. Conditional The conditional of the propositions p and q is the compound statement “If p, then q” denoted as 𝑝𝑝 → 𝑞𝑞 which is false only when both p is true, and q is false. Truth Table for Conditional 𝒑𝒑 𝒒𝒒 𝒑𝒑 → 𝒒𝒒 T T T T F F F T T F F T Example 3. Write each statement in “if-then” form. a. All right angles are equal. b. Equal quantities multiplied by equal quantities are equal. c. All triangles have three sides. Solution: a. If two angles are right angles, then they are equal. b. If equal quantities are multiplied by equal quantities, then the results are equal quantities. c. If a figure is a triangle, then it has three sides. The clause following the word “if” of a statement in “if-then” form is called the hypothesis of the statement and the clause following “then” is called the conclusion. Example 4. Identify the hypothesis and the conclusion in each statement. a. Opposite sides of a rectangle are parallel. b. Complements of the same angle are equal. Solution: a. If two sides are opposite sides of a rectangle, then they are parallel. Hypothesis: Two sides are opposite sides of a rectangle. Conclusion: The two sides are parallel. b. If two angles are complements of the same angle, then they are equal. Hypothesis: Two angles are complements of the same angle. Conclusion: The angles are equal. 5. Biconditional The biconditional of the propositions p and q is the compound statement “p if and only if q” denoted as 𝑝𝑝 ↔ 𝑞𝑞 which is false only when both p is true, and q is false. Truth Table for Biconditional 𝒑𝒑 𝒒𝒒 𝒑𝒑 ↔ 𝒒𝒒 T T T T F F F T F F F T Example 6. Determine if the following statements are true or false. a. 3 + 3 = 6 if and only if 2 + 3 = 5. b. 1 + 2 = 3 if and only if 3 + 4 = 9. Solution: a. This is a true statement since “3 + 3 = 6” and “2 + 3 = 5” are both true statements. b. This is false statement since “1 + 2 = 3” is true but “3 + 4 = 9” is false. Example: A. Construct the truth table for each of the following statements. 1. ~(~𝑝𝑝 ˅ 𝑞𝑞) ˅ 𝑞𝑞 𝒑𝒑 𝒒𝒒 ~𝒑𝒑 ~𝒑𝒑 ˅ 𝒒𝒒 ~(~𝒑𝒑 ˅~𝒒𝒒) ~(~𝒑𝒑 ˅~𝒒𝒒) v 𝒒𝒒 T T F T F T T F F F T T F T T T F T F F T T F F Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  5 2. (~𝑝𝑝 ˅ 𝑞𝑞) ˅ (𝑝𝑝 ˄ ~𝑞𝑞) 𝒑𝒑 𝒒𝒒 ~𝒑𝒑 ~𝒒𝒒 ~𝒑𝒑 ˅ 𝒒𝒒 𝒑𝒑 ˄ ~𝒒𝒒 (~𝒑𝒑 ˄ 𝒒𝒒)˅ (𝒑𝒑 ˄ ~𝒒𝒒) T T F F T F T T F F T F T T F T T F T F T F F T T T F T 3. (~𝑝𝑝 ˅ ~𝑞𝑞) → ~(𝑝𝑝 ˄ 𝑞𝑞) 𝒑𝒑 𝒒𝒒 ~𝒑𝒑 ~𝒒𝒒 (~𝒑𝒑 ˅ ~𝒒𝒒) (𝒑𝒑 ˄ 𝒒𝒒) ~(𝒑𝒑 ˄ 𝒒𝒒) (~𝒑𝒑 ˄ ~𝒒𝒒) → ~(𝒑𝒑 ˄ 𝒒𝒒) T T F F F T F T T F F T T F T T F T T F T F T T F F T T T F T T Now, we will discuss the conditional statements and how to convert this into its converse, inverse, and contrapositive form. The Conditional and Related Statement What are the equivalent forms of the Conditional? Every conditional statement can be stated in many equivalent forms. It is not even necessary to state the antecedent before the consequent. For instance, the conditional “If I live in Boston, then I must live in Massachusetts” can be also stated as “I must live in Massachusetts, if I live in Boston.” Common Forms of 𝒑𝒑 → 𝒒𝒒 Every conditional statement 𝒑𝒑 → 𝒒𝒒 can be written in the following equivalent forms. If p, then q Every p is a q. If p, q. q, if p. p only if q. q provided that p. p implies q q is a necessary condition for p. Not p or q p is a sufficient condition for q. Example: Write each of the following is “If p, then q” form. a. The number is an even number if it is divisible by 2. b. Today is Friday, only if yesterday was Thursday. Solution: a. The statement, “The number is an even number provided that it is divisible by 2.,” is in “q provided that p” form. The antecedent is “it is divisible by 2,” and the consequent is “the number is an even number.” thus its “If p, then q” form is “If it is divisible by 2, then the number is an even number.” b. The statement, “Today is Friday, only if yesterday was Thursday” is in “p only if q” form. The antecedent is “Today is Friday.” The consequent is “yesterday was Thursday.” Its “If p, then q” form is “If today is Friday, then yesterday was Thursday.” After discussing implication or conditional statement, we will now proceed with constructing the converse, inverse, and contrapositive of every conditional statement. Converse, Inverse and Contrapositive For every implication or conditional statement, you can construct its converse, inverse, and contrapositive. If the hypothesis and the conclusion in an implication are reversed, the new statement is called the converse of the given statement. In symbols, the converse of 𝑝𝑝 → 𝑞𝑞 is 𝒒𝒒 → 𝒑𝒑. Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  6 The inverse of an implication, “If 𝑝𝑝 then 𝑞𝑞” is “If not 𝑝𝑝, then not 𝑞𝑞.” In symbols, the invers of 𝑝𝑝 → 𝑞𝑞 is ~𝒑𝒑 → ~𝒒𝒒. The contrapositive of an implication is the converse of its inverse. To form the contrapositive of a statement, we interchange the hypothesis and the conclusion and negate each. In symbols, the contrapositive of 𝑝𝑝 → 𝑞𝑞 is ~𝒒𝒒 → ~𝒑𝒑. Example 1. Write the converse, inverse, and contrapositive of the statement below: a. The opposite angles of a parallelogram are equal. Solution: p: Two angles of a parallelogram are opposite angles. q: They are equal. Conditional (𝑝𝑝 → 𝑞𝑞): If two angles of a parallelogram are opposite angles, then they are equal. Converse (𝑞𝑞 → 𝑝𝑝): If two angles of a parallelogram are equal, then they are opposite angles. Inverse (~𝑝𝑝 → ~𝑞𝑞): If two angles of a parallelogram are not opposite angles, then they are unequal. Contrapositive (~𝑞𝑞 → ~𝑝𝑝): If two angles of a parallelogram are unequal, then they are not opposite angles. Want to know more? If you need to read or review, kindly access Intro to Truth Tables and Boolean Algebra (2017). https://medium.com/i-math/intro-to-truth-tables-boolean-algebra-73b331dd9b94 and watch the videoTruth Table – Discrete Mathematics Logic by Emily Jane. Reference list 1. Aufmann, R. N., Lookwood, J. S., Nation, R. D., & Clegg, D. K. (2013). Mathematical excursion. Brooks/ Cole. 2. Bautista, E., Cabral, E., Garces, I. J., Garciano, A., Muga, F., & de Lara-Tuprio, E. (2006). Introductory algebra. Vibal Publishing House, Inc. 3. Berry, B. (2019, October 8). Intro to truth tables & boolean algebra - math hacks. Medium. https://medium.com/i-math/intro-to-truth-tables-boolean-algebra-73b331dd9b94 4. Cengage Learning. (2018). Mathematics in the modern world. Rex Book Store, Inc. 5. David Lippman, Extended Learning Institute (ELI), Northern Virginia Community College. (n.d.). Truth tables and analyzing arguments: Examples | mathematics for the liberal Arts. Lumen Learning. https://courses.lumenlearning.com/math4libarts/chapter/truth-tables-and-analyzing-arguments-examples/ 6. Rodriguez, M. J., Salvador, I. G., Ragma, F., Torres, E., Manalang, E., Oredina, N., & Ogoy, J. (2018). Mathematics in the modern world. Nieme Publishing House Co. Ltd. Supplementary Videos 1. Emily S. (2014, March 26). Truth table tutorial - discrete mathematics logic [Video]. YouTube. https://www.youtube.com/watch?v=wRMC-ttjhwM&t=226s 2. Numberbender. (2019, February 7). Paano gumawa ng truth table mula sa compound statement [Video]. YouTube. https://www.youtube.com/watch?v=1LUOJqlu81c 3. TheTrevTutor. (2017, July 17). Introduction to propositional logic - discrete mathematics [Video]. YouTube. https://www.youtube.com/watch?v=itrXYg41-V0&t=274s 4. WOW MATH. (2020, December 2). Mathematics in the modern world [Video]. YouTube. https://www.youtube.com/playlist?list=PLPPsDIdbG32Auf61Nq_mFwIe7Xel3VfsW Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd