Introduction to Logics - MAT101 Finals - Lesson 3 PDF

Introduction to Logics MAT101: FINALS - Lesson 3  At the end of this lesson, the learners should be able to: 1. Define logic and its significance in mathematics and reasoning. 2. Identify different types of statements and their logical properties. 3. Apply logical connectives to const...

Introduction to Logics MAT101: FINALS - Lesson 3  At the end of this lesson, the learners should be able to: 1. Define logic and its significance in mathematics and reasoning. 2. Identify different types of statements and their logical properties. 3. Apply logical connectives to construct compound statements. 4. Analyze arguments for validity using truth tables and logical equivalences. 5. Solve problems involving logical reasoning.  What is Logic? Logic is the study of reasoning and the principles that govern valid arguments and sound decision- making. It is a foundational tool in mathematics, philosophy, computer science, and everyday problem-solving.  KEY CONCEPTS Proposition - a declarative sentence that is either true or false, but not both true and false - represented by small letters usually p, q, r and s. Truth Value – the value of a proposition that can be either True (T) or False (F) Consider the following statements. Identify if they are propositions or not. If yes, determine their truth value. 1) p = Manila is the capital of the Philippines. This sentence is a proposition since it is a declarative sentence and it is TRUE. 2) q = Dogs are mammals. Same as number 1, the sentence is also a declarative sentence and it is TRUE. 3) r = It is Tuesday? This sentence is NOT a proposition since it is not a declarative sentence. It is an interrogative sentence since it asks a question. 4) s = Go get the car. Since it gives a command, this is an imperative sentence, thus, NOT a proposition. 5) t = 8 is a prime number. The sentence is declarative, so it is a proposition. However, 8 is not a prime number so it has a “FALSE” truth value. 6) u = This is a nice car. The sentence is declarative, but we cannot determine its truth value (whether TRUE or FALSE). This is because it is subjective. A basic car may be nice to someone who is not familiar with different models while a car enthusiast may not find it as nice. Then, we can’t have a definite truth value when it comes to the sentence.  Types of Propositions 1) Simple Proposition – a statement that conveys a single idea. p = Jonathan likes to play video games. q = Jonathan always stay up late. 2) Compound Proposition – a statement that conveys two or more ideas. Jonathan likes to play video games and always stay up late.. The word “and” connects the two simple propositions to create a compound proposition. It is an example of logical connective. Logical Connectives 1) Negation - read as “not” and has a symbol ~ Consider the following propositions: p = Today is Sunday. q = The shop is closed. ~p: “Today is not Sunday” ~q: “The shop is not closed” Logical Connectives 2) Conjunction - read as “and” and has a symbol Λ Consider the following propositions: p = Today is Sunday. q = The shop is closed. p Λ q: “Today is Sunday and the shop is closed.” Logical Connectives 3) Disjunction - read as “or” and has a symbol V Consider the following propositions: p = Today is Sunday. q = The shop is closed. p V q: “Today is Sunday or the shop is closed.”  Logical Connectives 4) Implication - read as “if…else” and has a symbol → Consider the following propositions: p = Today is Sunday. q = The shop is closed. p → q: “If today is Sunday, then the shop is closed.”  Logical Connectives 5) Biconditional Statement - read as “if and only if” and has a symbol Consider the following propositions: p = Today is Sunday. q = The shop is closed. p q: “Today is Sunday if and only if the shop is closed.”  Summary of Logical Connectives Connectives Word Form Statement Symbolic Form Negation not not p ~p Conjunction and p and q pΛq Disjunction or p or q pVq Implication if…then if p… then q p→q Biconditional if and only if p if and only if q p q Statement  Example: Change the following propositions from symbol form to sentence form given: p = Joy watched the concert of Ben&Ben. q = Joy studies for the test. r = Joy is fully rested. s = Joy passed the test.  p = Joy watched the concert of Ben&Ben. Example: q = Joy studies for the test. r = Joy is fully rested. s = Joy passed the test. 1) p Λ q Joy watched the concert of Ben&Ben and Joy studies for the test. 2) q V r Joy studies for the test or she is fully rested. Note: We may use pronouns (she in this case) or change some structure of the sentence as long as it delivers the intended thought of the logical connectives.  p = Joy watched the concert of Ben&Ben. Example: q = Joy studies for the test. r = Joy is fully rested. s = Joy passed the test. 3) ~q Joy does not study for the test. 4) q→ s If Joy studies for the test, then she will pass the test.  p = Joy watched the concert of Ben&Ben. Example: q = Joy studies for the test. r = Joy is fully rested. s = Joy passed the test. 5) s (q Λ r) Joy passed the test if and only if she studies for the test and is fully rested. 6) (~p Λ r) → (q Λ s) If Joy did not watch the concert of Ben&Ben and is fully rested, then she studies for the test and passed the test.  Truth Table - a diagram in rows and columns used to organize, present, and show the relationships between the truth values of propositions involving logical connectives. The letter “T” in the truth table represents the truth value TRUE while the letter “F” represents FALSE. As a side note, other books are using 1 for TRUE and 0 for FALSE. p q pVq (p V q) Λ p T T T T T F T T F T T F F F F F Notice that the values of p and q were all the possible combinations of TRUE and FALSE given the two propositions. Then the truth values were evaluated based on the compound proposition being solved. Tautology - a compound proposition is said to be a tautology if it generates a TRUE truth value in all the possible combinations of the simple propositions. Contradiction - a compound proposition is said to be a contradiction if it generates a FALSE truth value in all the possible combinations of the simple propositions. Contingency - a proposition that is neither a tautology nor a contradiction.  Truth Values of Logical Propositions 1) Negation - equal to the opposite truth value of the proposition p ~p Given proposition p: T F F T  Truth Values of Logical Propositions 2) Conjunction - only has a “TRUE” truth value when both of the propositions are “TRUE”. Otherwise, it has a “FALSE” truth value p q pΛq Given propositions p and q: T T T T F F F T F F F F  Truth Values of Logical Propositions 3) Disjunction - has a “TRUE” truth value when either one or both of the propositions has a “TRUE” truth value. Otherwise, it has a “FALSE” truth value p q pVq Given propositions p and q: T T T T F T F T T F F F  Truth Values of Logical Propositions 4) Implication - has a “FALSE” truth value when the first proposition is “TRUE” and the second proposition is “FALSE”. Otherwise, it has a “TRUE” truth value p q p→q Given propositions p and q: T T T T F F F T T F F T  Truth Values of Logical Propositions 5) Biconditional Statement - has a “TRUE” truth value when both propositions share the same truth value. Otherwise, it has a “FALSE” truth value p q p q Given propositions p and q: T T T T F F F T F F F T Example: Create a truth table given the following compound propositions p, q, and r: 1) ∼ p ∧ q p q ~p ∼p∧ q Conjunction T T F F - only has a “TRUE” truth value when both of the T F F F propositions are “TRUE”. Otherwise, it has a F T T T “FALSE” truth value F F T F  Example: Implication - has a “FALSE” truth value when the first 2) (p → q) ∼q proposition is “TRUE” and the second proposition is “FALSE”. Otherwise, it has a “TRUE” truth p q p→q ∼q (p → q) ∼q value T T T F F Biconditional Statement - has a “TRUE” truth value T F F T F when both propositions share F T T F F the same truth value. Otherwise, it has a “FALSE” F F T T T truth value  Example: p q r p∨q (p ∨ q) ∧ r 3) (p ∨ q) ∧ r T T T T T T T F T F For this example, since we have T F T T T three propositions (p, q, r), we T F F F F have to consider all of the possible F T T T T combinations of their truth values. This is why we have eight (8) F T F T F values for p, q and r. F F T F F F F F F F  Example: p q r p∨q (p ∨ q) ∧ r 3) (p ∨ q) ∧ r T T T T T Conjunction - only has a “TRUE” truth T T F T F value when both of the propositions are T F T T T “TRUE”. Otherwise, it has a “FALSE” truth T F F F F value Disjunction - has a “TRUE” truth value F T T T T when either one or both of the F T F T F propositions has a “TRUE” truth value. F F T F F Otherwise, it has a “FALSE” truth value F F F F F THANK YOU!