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- 1. Logic Is the study of reasoning; it is specifically concerned with whether reasoning is correct. It focuses on the relationship among statements as opposed to the content of any particular statement. Proposition A sentence that is either true or false, but not both. It is typically expressed as a declarative sentence (as opposed to a question, command, etc.). Propositions are the basic building blocks of any theory of logic.
- 2. Which sentences are proposition? 1. The only positive integers that divide 7 are 1 and 7 itself. 2. Alfred Hitchcock won an Academy Award in 1940 for directing “Rebecca.” 3. For every positive integer n, there is a prime number larger than n. 4. Earth is the only planet in the universe that contains life. 5. Buy two tickets to the “Never Say Never” concert.
- 3. We will use variables, such as p, q, and r, to represent propositions. Let p and q be propositions. The conjunction of p and q, denoted p ʌ q, is the proposition p and q The disjunction of p and q, denoted p v q, is the proposition p or q
- 4. example: if p: It is raining, q: It is cold, then the conjunction of p and q is p ʌ q: It is raining and It is cold. The disjunction of p and q is p v q: It is raining or it is cold. The inclusive-or of propositions p and q is true if both p and q are true.
- 5. The exclusive-or that defines p exor q to be false if both p and q are true. Exclusive-or p T T F F q T F T F p exor q F T T F
- 6. Truth tables The truth values of propositions such as conjunctions and disjunctions can be described by truth tables. Conjunction Disjunction pʌ q T T T T F F F T F F F F The binary operator on a set elements in X an element of X. p q p T T F F q T F T F pvq T T T F X assigns to each pair of ʌ and v are both binary operator on propositions.
- 7. The negation of p, denoted by ¬p, is the proposition not p example: p: Paris is the capital of England. ¬p: Paris is not the capital of England. Negation p T F ¬p F T A unary operator on a set X assigns to each element in X an element of X. ¬ is a unary operator on propositions.
- 8. Operator Precedence In expressions involving some or all of the operators ¬, ʌ, and v, in the absence of parentheses, we first evaluate ¬, then ʌ, and then v. example: Given that proposition p is false, proposition q is true, and proposition r is false, determine whether the proposition ¬p v q ʌ r is true or false.
- 9. Exercises: I.Given that proposition p is false, proposition q is true, and proposition r is false, determine whether each proposition is true or false. 1. p v q 3. ¬p v q 5. ¬(p v q) ʌ (¬p v r) II. 2. ¬p v ¬q 4. ¬p v ¬(q ʌ r) 6. (p v ¬r) ʌ ¬[(q v r) v ¬(r v p)] Write the truth table of each proposition. 7. p ʌ ¬q 9. (p v q) ʌ ¬p 11. (p ʌ q) v (¬p v q) 13. ¬(p ʌ q) v (¬q v r) 8. (¬p v ¬q) v p 10. (p ʌ q) ʌ ¬p 12. ¬(p ʌ q) v (r ʌ ¬p) 14. (p ʌ q) ʌ ¬(r v p)
- 10. III. Represent the proposition symbolically by letting p: There is a storm. q: It is raining. 15. There is no storm. 16. There is a storm and it is raining. 17. There is a storm, and it is not raining. 18. There is no storm and it is not raining. 19. Either there is a storm or it is raining. 20. Either there is a storm or it is raining, but there is no storm.
- 11. Conditionals The implication (or conditional) p → q, is the proposition “if p, then q”, that is false when p is true and q is false and true otherwise. In this implication p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence). Implication p T T F F q T F T F p →q T F T T
- 12. Other ways of expressing implication: “p implies q” “p is sufficient for q” “when p then q” “q if p” “if p, q” “q whenever p” “p only if q” “q is necessary for p” Example: Rewrite the given implication in other forms. If it snows, then the streets are slick. 1.The streets are slick if it snows. 2.Snowy weather implies the streets are slick. 3.When it snows, then the streets are slick. 4.It snows only if the streets are slick.
- 13. Related Conditionals The converse of the implication “if p, then q”, is the implication “if q, then p.” That is, the converse of p→q is q→p. The inverse of the implication “if p, then q”, is the implication “if not p, then not q” that results when p and q are replaced by their negations; that is, the inverse of p→q is ¬p→¬q The contrapositive of the implication “if p, then q”, is the implication “if not q, then not p.” That is, the contrapositive of p→q is ¬q→¬p.
- 14. Example: Find the converse, inverse, contrapositive of the given conditional. and If it rains, then I buy a new umbrella. Converse: If I buy a new umbrella, then it is raining. Inverse: If it does not rain, then I do not buy a new umbrella. Contrapositive: If I do not buy a new umbrella, then it is not raining.
- 15. The biconditional p↔q is the proposition that is true when p and q have the same truth values and is false otherwise. “p if and only if q” “p is necessary and sufficient for q” “if p then q, and conversely” Biconditional p T T F F q T F T F p ↔q T F F T
- 16. Propositional Equivalences Tautology is a compound proposition that is always true, no matter what the truth values of the propositions that occur in it. Contradiction is a compound proposition that is always false. Contingency is a proposition that is neither a tautology nor a contradiction. Example of Tautology and Contradiction p T F ¬p F T p v ¬p T T p ʌ ¬p F F
- 17. Logical Equivalence Compound propositions that have the same truth values in all positive cases are called logically equivalent. The propositions p and q are called logically equivalent if p↔q is a tautology. The notation p⇔q denotes that p and q are logically equivalent. Ex. Show that the propositions p v (q ʌ r) and (p v q) ʌ (p v r) are logically equivalent. Table
- 18. Exercises: I.Let A: “It is snowing” B: “The roofs are white” C: “The streets are slick” D: “The trees are covered with ice” Write the following in symbolic notation. 1. If it is snowing, then the trees are covered with ice. 2. If it is not snowing, then the roofs are not white. 3. If the streets are not slick, then it is not snowing. 4. If the streets are slick, then the trees are not covered with ice.
- 19. II. State the converse, inverse, and contrapositive of each of the following conditionals. 1. If a triangle is a right triangle, then one angle has a measure of 900. 2. If a number is a prime, then it is odd. 3. If two lines are parallel, then the alternate interior angles are equal. 4. If Joyce is smiling, then she is happy. III. Using truth tables, determine whether the following pairs of statements are equivalent. 1. p v q; ¬p→q 2. ¬(p ʌ q); ¬p v ¬q 3. ¬(p v q); ¬p ʌ ¬q 4. p→q; ¬p→¬q
- 20. IV. Using truth tables, determine if the argument is tautology, contradiction or contingency. 1.¬p → (p→q) 2.[p ʌ (p→q)] → ¬q 3.p ʌ ¬p 4.[(p→q) ʌ (q→r)] → (p→r) 5.[(p v q) ʌ (p→r) ʌ (q→r)] → r 6.(p→q) ʌ (p ʌ ¬q)
- 21. Quantifiers, Venn Diagrams, and Valid Arguments Quantifiers give information about “how many” in the statements in which they occur. ex. all, some, no Quantified Statements are statements involving quantifiers. ex. Some women have red hair. All bananas are yellow. No professors are bald-headed.
- 22. Universal Quantifiers ( ∀ ) The words all, every, and each are called universal quantifiers because when these words are added to an open sentence to make it a statement, the sentence must be true in all possible instances in order for the statement to be true. ∀xP(x) “for all x P(x)” or “for every x P(x) examples: All prime numbers greater than 2 are odd. Every automobile pollutes the atmosphere. For each number x, x + 3 = 3 + x.
- 23. Existential Quantifiers ( ∃ ) Other quantified statements are intended to indicate that there exists at least one case in which the statement is true. Such statements generally involve one of the existential quantifiers: some, there exist, or there exists at least one. ∃xP(x) “there is an x such that P(x)” or “there is at least one x such that P(x) examples: Some men have black hair. There exist students who do not work hard. There exists at least one student who does not work hard.
- 24. Venn Diagrams A diagram in which the interiors of simple closed curves such as circles are used to represent collections of objects (or sets). It provide geometrical representations of the relationships indicated by statements involving quantifiers. ex. Draw Venn diagrams for the statements. 1.All dogs (D) are animals (A). 2.Some kindergarten students (K) are able to read (R). 3.No cat (C) is a dog (D).
- 25. Valid Arguments The argument is valid if in all cases where the premises are true, the conclusion is true. Ex. Consider the following two arguments. Use Venn diagrams to determine whether either is valid. a)All leghorns are chickens. All chickens are fowls. Therefore, all leghorns are fowls. b)All leghorns are chickens. All chickens are fowls. Therefore, all fowls are leghorns.
- 26. If, in some case, the conjunction of the premises is true and the conclusion is false, then the argument is invalid. Invalid arguments are sometimes called fallacies. Theorem is a statement that can shown to be true. Proof is a sequence of statements that form an argument to demonstrate that a theorem is true. Axioms or Postulates are the underlying assumptions about mathematical structures, the hypotheses of the theorem to be proved, and previously proved theorem.
- 27. Lemma is a simple theorem used in the proof of other theorems. Corollary is a proposition that can be established directly from a theorem that has been proved. Conjecture is a statement whose truth value is unknown. When a proof of a conjecture is found, the conjecture becomes a theorem. Many times conjectures are shown to be false, so they are not theorems. Rules of Inference which are means used to draw conclusions from other assertions, tie together the steps of a proof.
- 28. Rules of Inference Rule Tautology Name p ∴p∨q p → ( p ∨ q) Addition p∧q ∴p ( p ∧ q) → p Simplification p q ∴p∧q ( ( p) ∧ ( q) ) → ( p ∧ q) Conjuction
- 29. Rule p p→q ∴q ¬q p→q ∴ ¬p p→q q→r ∴p→r p∨q ¬p ∴q Tautology [ p ∧ ( p → q)] → q [ ¬q ∧ ( p → q ) ] → ¬p Name Modus ponens Modus tollens [( p → q) ∧ ( q → r )] → ( p → r ) Hypothetical syllogisms or chain rule [ ( p ∨ q ) ∧ ¬p ] → q Disjunctive syllogisms

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