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Understanding Logic: Propositions, Compound Propositions, and Truth Tables

This chapter explores the principles of logic, specifically focusing on propositions, compound propositions, conjunction, disjunction, and negation. Logic is essential in fields like mathematics and computer science as it helps establish correct reasoning and validate arguments. Propositions are statements that can be classified as either true or false, while compound propositions are formed by combining propositions using logical operators. This chapter also covers truth tables, which visually represent the truth values of compound propositions based on their components.

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Understanding Logic: Propositions, Compound Propositions, and Truth Tables

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  1. L 2 Chapter 1 Logic and Proofs

  2. Topics • Logic • Proposition • Compound proposition • Conjunction (AND) • Dis conjunction (OR) • And TRUE Table • Or TRUE Table • Not TRUE Table

  3. Logic • The study of reasoning • Concerned with whether reasoning is correct or not. • Focuses on relationship as opposed to content of any particular statement. • Logic is used in math to prove Thermo. • Logic is used in computer programs to prove if a program does what it supposed to do.

  4. Logic • When logic is used in math, a general method of proof is used called math induction. • Used throughout math and computer Science specially in discrete math.

  5. Proposition (p) • A statement that is either True or False but not both. • Examples: • The Class starts at 12:00 • Jackie is over than 22 years old. • We will use p, q, r to perform propositions

  6. Proposition • Proposition is a declarative statement not an order or a command. • Joe is single  a proposition • Is Joe single?  NOT a proposition • Are you single?  NOT a proposition. • Do you homework. NOT a proposition • P: 1 + 4 = 10 T or F • P: it is cold T or F • q: it is raining T or F

  7. Compound Proposition • In ordinary speeches we use the words AND & OR • p AND q is denoted as p ^ q • p or q is denoted as p v q Proposition resulted from combining the above propositions is called compound proposition.

  8. Compound position • P: It is Cold and Raining (Cold ^ Raining) • Possibilities: • Cold and also Raining • Cold but not Raining • Not Cold but Raining • Not Cold and Not Raining • Same logic and similar propositions for (OR V) • P: It is Cold or Raining (Cold V Raining)

  9. Conjunction and Disconjunction • Conjunction: • p and q • p && q • p ^ q • Disconjunction: • p or q • p || q • p V q

  10. Negation (Not) • Denoted by the symbol ¬ • NOT p • ¬ p • ¬ q

  11. The truth table for the compound proposition p^ q And truth p ^ q True is denoted by 1 and False is denoted by 0 If P: 5 + 6 = 10 q =7 Then p ^ q is F

  12. The truth table for the compound proposition pV q And truth p V q True is denoted by 1 and False is denoted by 0 If P: 5 + 6 = 10 q =7 Then p V q is T

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