Slide subtopic 4
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The document discusses tautologies and contingency tables. It provides examples of a tautology and how a contingency table classifies subjects by two variables into cells. It also defines arguments and their validity, listing several common rules of inference like modus ponens, modus tollens, disjunctive syllogism. Examples are given of representing arguments symbolically and determining their validity. Indirect proofs via contradiction are defined as assuming the negation of the conclusion and deriving the negation of a premise, resulting in a contradiction.
![p→q
Determine whether the argument
p
∴q
Is valid
[First solution] We construct a
truth table for all the propositions
involved.
P q p→q p q
T T T T T
T F F T F
F T T F T
F F T F F](https://image.slidesharecdn.com/slidesubtopic4-120520092207-phpapp02/85/Slide-subtopic-4-11-320.jpg)
![This method is based on Modus Ponens,
[(p ⇒ q) ˄ p ] ⇒ q
Virtually all mathematical theorems are
composed of implication of the type,
(
The are called the hypothesis or premise, and
q is called conclusion. To prove a theorem
means to show the implication is a tautology.
If all the are true, the q must be also true.](https://image.slidesharecdn.com/slidesubtopic4-120520092207-phpapp02/85/Slide-subtopic-4-19-320.jpg)
Slide subtopic 4
- 1. UNIVERSITI PENDIDKAN SULTAN IDRIS PREPARED BY : MOHAMAD AL FAIZ BIN SELAMAT
- 3. Tautology: A proposition that is always true for all possible value of its propositional variables. Example of a Tautology The compound proposition p ˅¬p is a tautology because it is always true. P ¬p p ˅ ¬p T F T F T T
- 5. P ¬p p ˅ ¬p T F F F T F
- 7. A contingency table is a table of counts. A two-dimensional contingency table is formed by classifying subjects by two variables. One variable determines the row categories; the other variable defines the column categories. The combinations of row and column categories are called cells. Examples include classifying subjects by sex (male/female) and smoking status (current/former/never) or by "type of prenatal care" and "whether the birth required a neonatal ICU" (yes/no). For the 1. mathematician, a two-dimensional contingency table with r rows and c columns is the set {xi j: i =1... r; j=1... c}.
- 9. : , Definition: An argument is a sequence of propositions written The symbol ∴ is read “therefore.” The propositions, ,… are called the hypotheses (or premises) or the proposition q is called the conclusion. The argument is valid provide that if the proposition are all true, then q must also be true; otherwise, the argument is invalid (or a fallacy).
- 11. p→q Determine whether the argument p ∴q Is valid [First solution] We construct a truth table for all the propositions involved. P q p→q p q T T T T T T F F T F F T T F T F F T F F
- 13. Rule of inference Name p→q Modus ponens p ∴q p→q Modus tollens ⌐q ∴ ⌐p Addition p ∴p˅q Simplification p˅q ∴p p Conjunction q ∴p˅q p→q Hypothetical syllogism q→r ∴p→r p˅q Disjunctive syllogism ⌐p ∴q
- 15. Represent the argument. The bug is either in module 17 or in module 81 The bug is a numerical error Module 81 has no numerical error ___________________________________________ ∴ the bug is in module 17. Given the beginning of this section symbolically and show that it is valid. If we let p : the bug is in module 17. q : the bug is in module 81. r : the bug is numerical error.
- 17. The argument maybe written pVq r r → ⌐q ∴p From r → ⌐q and r, we may use modus ponens to conclude ⌐q. From r V q and ⌐q, we may use the disjunctive syllogism to conclude p. Thus the conclusion p follows from the hypotheses and the argument is valid.
- 19. This method is based on Modus Ponens, [(p ⇒ q) ˄ p ] ⇒ q Virtually all mathematical theorems are composed of implication of the type, ( The are called the hypothesis or premise, and q is called conclusion. To prove a theorem means to show the implication is a tautology. If all the are true, the q must be also true.
- 21. Solution: Let p: x is odd, and q: x2 is odd. We want to prove p → q. Start: p: x is odd → x = 2n + 1 for some integer n → x2 = (2n + 1)2 → x2 = 4n2 + 4n + 1 → x2 = 2(2n2 + 2n) + 1 → x2 = 2m + 1, where m = (2n2 + 2n) is an integer → x2 is odd → q
- 23. Definition: An indirect proof uses rules of inference on the negation of the conclusion and on some of the premises to derive the negation of a premise. This result is called a contradiction. Contradiction: to prove a conditional proposition p ⇒ q by contradiction, we first assume that the hypothesis p is true and the conclusion is false (p˄ ~ q). We then use the steps from the proof of ~q ⇒ ~p to show that ~p is true. This leads to a contradiction (p˄ ~ p), which complete the proof.
- 25. Proof: Assume that x is even (negation of conclusion). Say x = 2n (definition of even). Then = (substitution) = 2n · 2n (definition of exponentiation) = 2 · 2n2 (commutatively of multiplication.) Which is an even number (definition of even) This contradicts the premise that is odd.