Discrete Structures

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## Discrete Structures

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**Discrete Structures**Chapter 3: The Logic of Quantified Statements 3.4 Arguments with Quantified Statements The only complete safeguard against reasoning ill, is the habit of reasoning well; familiarity with the principles of correct reasoning; and practice in applying those principles. – John Stuart Mill, 1806 – 1873 3.4 Arguments with Quantified Statements**Universal Instantiation (in-stan-she-AY-shun)**• The rule of universal instantiation is that if some property is true of everything in a set, then it is true of any particular thing in the set. 3.4 Arguments with Quantified Statements**Universal Instantiation**• One of the most famous examples of universal instantiation is as follows: • All men are mortal. • Socrates is a man. • Socrates is mortal. 3.4 Arguments with Quantified Statements**Universal Modus Ponens**• Universal modus ponens is a combination of universal instantiation and modus ponens. It is Latin for "mode that affirms". Formal VersionInformal Version • The first line is called the major premiseand the second line is the minor premise. 3.4 Arguments with Quantified Statements**Example – pg. 142 # 2**• Use universal instantiation or universal modus ponens to fill in valid conclusions. • If an integer n equals 2k and k is an integer, then n is even. • 0 equals 20 and 0 is an integer. • . 3.4 Arguments with Quantified Statements**Example – pg. 142 # 3**• Use universal instantiation or universal modus ponens to fill in valid conclusions. • For all real numbers a, b, c, and d, if b 0 and d 0, then a/b + c/d = (ad + bc)/(bd). • a = 2, b = 3, c = 4, and d = 5 are particular real numbers such that b 0 and d 0. • . 3.4 Arguments with Quantified Statements**Definition**• Valid • To say that an argument form is valid means the following: No matter what particular predicates are substituted for the predicate symbols in its premises, if the resulting premise statements are all true, then the conclusion is also true. An argument is called validiff its form is valid. 3.4 Arguments with Quantified Statements**Using Diagrams to Show Validity**• Example • (informal) All integers are rational numbers. • (formal) ∀integers n, n is a rational number. 3.4 Arguments with Quantified Statements**Using Diagrams to Show Invalidity**• Show invalidity of the following argument • All human beings are mortal. • Felix is mortal. • ∴ Felix is a human being. 3.4 Arguments with Quantified Statements**Modus Ponens in Diagrams**3.4 Arguments with Quantified Statements**Example – pg. 143 # 21**• Indicated whether the arguments are valid or invalid. Support your answers by drawing diagrams. • All people are mice. • All mice are mortal. • All people are mortal. 3.4 Arguments with Quantified Statements**Universal Modus Tollens**• Universal modus ponens is a combination of universal instantiation and modus tollens. It is Latin for "mode that denies". Formal VersionInformal Version 3.4 Arguments with Quantified Statements**Example – pg. 142 # 5**• Use universal modus tollens to fill in valid conclusions for the arguments. • All irrational numbers are real numbers. • 1/0 is not a real number. • . 3.4 Arguments with Quantified Statements**Converse Error**• This claim is most simply put as • If A, then B. • B. • A. • It's a fallacy because at no point is it shown that A is the only possible cause of B; therefore, even if B is true, A can still be false. 3.4 Arguments with Quantified Statements**Converse Error**• In other words, a universal conditional statement is not logically equivalent to its converse. Formal VersionInformal Version Invalid conclusion 3.4 Arguments with Quantified Statements**Converse Error - Example**• If my car was Ferrari, it would be able to travel at over a hundred miles per hour. • I clocked my car at 101 miles per hour. • Therefore, my car is a Ferrari. 3.4 Arguments with Quantified Statements**Inverse Error**• The conditional statement is not logically equivalent to its inverse. Formal VersionInformal Version 3.4 Arguments with Quantified Statements**Inverse Error - Example**• If I hit my professor with a cream pie, she will flunk me. • I will not hit my professor with a cream pie. • Therefore, she will not flunk me. 3.4 Arguments with Quantified Statements**Example – pg. 143 # 15**• The argument may be valid by universal modus ponens or universal modus tollens; or the argument is invalid and exhibit the converse or inverse error. State which are valid and which are invalid. Justify your answers. • Any sum of two rational numbers is rational. • The sum r + s is rational. • The numbers r and s are both rational. 3.4 Arguments with Quantified Statements**Example – pg. 143 # 18**• The argument may be valid by universal modus ponens or universal modus tollens; or the argument is invalid and exhibit the converse or inverse error. State which are valid and which are invalid. Justify your answers. • If an infinite series converges, then the terms go to 0. • The terms of the infinite series do not go to 0. • The infinite series does not converge. 3.4 Arguments with Quantified Statements