Logic worksheet
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This document provides examples of truth tables for basic logical connectives such as negation, conjunction, disjunction, biconditional, and conditional. It explains that truth tables consider all possible cases for statements connected by logical operators. Several practice problems are included to have the student complete truth tables and identify tautologies, contradictions, and logically equivalent statements using truth tables.
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= P 4,800
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Interest = Principal x (1 + Rate)^Time - Principal
= P 60,000 x (1 + 0.08)^1 - P 60,000
= P 60,000 x 1.08 - P 60,000
= P 4,800
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This document introduces logical operators and compound statements. It defines logical operators as words or symbols that combine statements to form new statements. The four main logical operators discussed are conjunction ("and" symbolized as ^), disjunction ("or" symbolized as _), negation ("not" symbolized as :), and conditional ("if...then" symbolized as !). Truth tables are provided to show the truth values of statements combined with these operators. The document also discusses tautologies, contradictions, and the biconditional operator ("if and only if" symbolized as $).
Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكرو
Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.
Propositional Logic, Boolean Arithmetic And Binary Arithmetic
1. The document discusses the relationships between propositional logic, Boolean algebra, and binary arithmetic.
2. It presents logic tables to express logical connectives like negation, conjunction, disjunction using only negation and conjunction. Similarly, it constructs tables using set operations and binary operations on 0s and 1s.
3. The goal is to show that logical operations can be expressed in terms of more basic ones, and that propositional logic relates to other branches of mathematics like sets and binary arithmetic.
Propositional Logic, Boolean Arithmetic And Binary Arithmetic
1. The document discusses propositional logic, boolean algebra, and binary arithmetic. It presents truth tables using only negation and conjunction to define other logical operations like disjunction and implication.
2. Sets are also discussed, with comparisons made between logical operations and set operations using intersections and complements. Binary arithmetic is similarly compared using 1-r and r*s operations.
3. Further "extreme cases" are explored by defining new operations based on negation, conjunction, and the existing operations, producing complete tautology tables for logic, sets, and binary arithmetic. Equivalences between the systems are summarized.
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Logic worksheet
- 1. Name ________________________ Worksheet 1.2 Logic and Truth Tables We can represent the truth of expressions in a tabular form called “truth tables.” These tables consider all cases and can add great insight into otherwise complicated expressions. Here are examples of some of most basic truth tables. The truth table for negation (“not”) p ¬ p T F F T The truth table for disjunction (“or”) p q p q T T T T F T F T T F F F 1. Complete the following truth tables (a) Truth table for conjunction (“and”) p q p q T T T F F T F F (b) Truth table for biconditional (“if and only if”) p q p ↔ q T T T F F T F F Why do we say that the biconditional defines “logical equivalence”?
- 2. 2. The truth table for conditional (“if...then...”) can be quite confusing. Here it is. p q p → q T T T T F F F T T F F T Consider the example “For every integer n, if n > 2 then n2 > 4.” In this case p represents “n > 2” and q represents “n2 > 4.” Thus if n = 3 then p is true and q is true. (a) Find another value of n that makes p true and q true. (b) Find a value for n that makes p false and q false. (c) Find a value for n that makes p false and q true. (d) Find a value for n that make p true and q false. Explain any problems you have. Why are you having these problems? The third row, corresponding to part (c), frustrates many people—you are not alone. The truth has a special name. It is called “vacuously true.” 3. Complete the following truth tables (a) p q (¬ p) ↔ q T T T F F T F F (b) p q (¬ p) q T T T F F T F F (c) p q r q → r p → (q → r) T T T T T F T F T T F F F T T F T F F F T F F F
- 3. 4. Draw up and complete the truth tables for the following expressions. Please follow all of the conventions that you have observed above (such as listing p before q, etc.) (a) p (¬ p) (b) (¬ p) (¬ q) 5. An expression is a tautology if it is always true, regardless the truth of its component statements. Thus, the last column in its truth table will be all T’s. A statement is a contradiction if the last column is always F—it is never true. (a) Show that (p q) → p is a tautology by completing this truth table. p q p q (p q) → p T T T T T F T F F T F T F F F F (b) Show that p (¬ p) is a contradiction. p p (¬ p) T F
- 4. 6. Logically equivalent expressions have the same truth tables. Show that the following pairs of statements are logically equivalent. (a) (¬ p) q and (p q) (You saw this on the previous worksheet. Where?) (b) p → q and (¬ q) → (¬ p) (The second statement is called the Contrapositive) (c) (p q) r and (p r) (q r) (Distribution of sorts) (d) ¬ (p q) and (¬ p) (¬ q) (This, and part (e) is called De Morgan’s Law) (e) ¬ (p q) and (¬ p) (¬ q)