From 77a6fac5a6e607d1744c86b09ad2e94c6cf19a06 Mon Sep 17 00:00:00 2001 From: tomit4 Date: Sat, 6 Jun 2026 00:40:18 -0700 Subject: [PATCH] :construction: Setup for 3.4 --- chapter_3/exercises.md | 393 +++++++++++++++++++++++++++++++++++++ chapter_3/notes.md | 133 +++++++++++++ chapter_3/test_yourself.md | 25 +++ 3 files changed, 551 insertions(+) diff --git a/chapter_3/exercises.md b/chapter_3/exercises.md index 2207be1..5c5c2bf 100644 --- a/chapter_3/exercises.md +++ b/chapter_3/exercises.md @@ -2613,3 +2613,396 @@ $X = g$ c. $?\text{isabove}(g, X)$ $X = b_1, X = w_1$ + +--- + +**Exercise Set 3.4** + +Page 179 + +1. Let the following law of algebra be the first statement of an argument: For + all real numbers $a$ and $b$, + +$$ (a + b)^2 = a^2 + 2ab + b^2 $$ + +Suppose each of the following statements is, in turn, the second statement of +the argument. Use universal instantiation or universal modus ponens to write the +conclusion that follows in each case. + +a. $a = x$ and $b = y$ are particular real numbers. + +b. $a = f_i$ and $b = f_i$ are particular real numbers. + +c. $a = 3u$ and $b = 5v$ are particular real numbers. + +d. $a = g(r)$ and $b = g(s)$ are particular real numbers. + +e. $a = \log(t_1)$ and $b = \log(t_2)$ are particular real numbers. + +Use universal instantiation or universal modus ponens to fill in valid +conclusions for the arguments in 2-4. + +2. + +$$ +\text{If an integer} n \text{ equals } 2 \cdot k \text{ and } k \text{ is an integer, then } n \text{ is even.} \\ +0 \text{ equals } 2 \cdot 0 \text{ and } 0 \text{ is an integer.} \\ +\therefore \text{\_\_\_\_\_\_} +$$ + +3. + +$$ +\text{For all real numbers } a, b, c, \text{ and } d, \text{ if } b \neq 0 \text{ and } d \neq 0 \text{ then } \frac{a}{b} + \frac{c}{d} = \frac{(ad + bc)}{bd} \\ +a = 2, b = 3, c = 4, \text{ and } d = 5 \text{ are particular real numbers such that } b \neq 0 \text{ and } d \neq 0 \\ +\therefore \text{\_\_\_\_\_\_} +$$ + +4. + +$$ +\forall \text{ real numbers } r, a, \text{ and } b, \text{ if } r \text{ is positive, then } (r^q)^b = r^{ab} \\ +r = 3, a = \frac{1}{2}, \text{ and } b = 6 \text{ are particular real numbers such that } r \text{ is positive.} \\ +\therefore \text{\_\_\_\_\_\_} +$$ + +Use universal modus tollens to fill in valid conclusions for the arguments in 5 +and 6. + +5. + +$$ +\text{All irrational numbers are real numbers.} \\ +\frac{1}{0} \text{ is not a real number.} \\ +\therefore \text{\_\_\_\_\_\_} +$$ + +6. + +$$ +\text{If a computer program is correct, then compilation of the program does not produce error messages.} \\ +\text{Compilation of this program produces error messages.} \\ +\therefore \text{\_\_\_\_\_\_} +$$ + +Some of the arguments in 7-18 are valid by universal modus ponens or universal +modus tollens; others are invalid and exhibit the converse or the inverse error. +State which are valid and which are invalid. Justify your answers. + +7. + +$$ +\text{All healthy people eat an apple a day.} \\ +\text{Keisha eats an apple a day.} \\ +\therefore \text{Keisha is a healthy person.} +$$ + +8. + +$$ +\text{All freshmen must take a writing course.} \\ +\text{Caroline is a freshman.} \\ +\therefore \text{Caroline must take a writing course.} +$$ + +9. + +$$ +\text{If a graph has no edges, then it has a vertex of degree zero.} \\ +\text{This graph has at least one edge.} \\ +\therefore \text{This graph does not have a vertex of degree zero.} +$$ + +10. + +$$ +\text{If a product of two numbers is } 0 \text{, then at least one of the numbers is } 0 \text{.} \\ +\text{For a particular number } x \text{, neither } (2x + 1) \text{ nor } (x - 7) \text{ equals } 0 \text{.} \\ +\therefore \text{The product } (2x + 1)(x - 7) \text{ is not } 0 \text{.} +$$ + +11. + +$$ +\text{All cheaters sit in the back row.} \\ +\text{Monty sits in the back row.} \\ +\therefore \text{Monty is a cheater.} +$$ + +12. + +$$ +\text{If an 8-bit two's complement represents a positive integer, then the 8-bit two's complement starts with a 0.} \\ +\text{The 8-bit two's complement for this integer does not start with 0.} \\ +\therefore \text{This integer is not positive.} +$$ + +13. + +$$ +\text{For every student } x \text{, if } x \text{ studies discrete mathematics, then } x \text{ is good at logic.} \\ +\text{Tarik studies discrete mathematics.} \\ +\therefore \text{Tarik is good at logic.} +$$ + +14. + +$$ +\text{If compilation of a computer program produces error messages, then the program is not correct.} \\ +\text{Compilation of this program does not produce error messages.} \\ +\therefore \text{This program is correct.} +$$ + +15. + +$$ +\text{Any sum of two rational numbers is irrational.} \\ +\text{The sum } r + s \text{ is rational.} \\ +\therefore \text{The numbers } r \text{ and } s \text{ are both rational.} +$$ + +16. + +$$ +\text{If a number is even, then twice that number is even.} \\ +\text{The number } 2n \text{ is even, for a particular number } n \text{.} \\ +\therefore \text{The particular number } n \text{ is even.} +$$ + +17. + +$$ +\text{If an infinite series converges, then the terms go to 0.} \\ +\text{The terms of the infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ go to 0.} \\ +\therefore \text{The infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ converges.} +$$ + +18. + +$$ +\text{If an infinite series converges, then the terms go to 0.} \\ +\text{The terms of the infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ do not go to 0.} \\ +\therefore \text{The infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ does not converge.} +$$ + +19. Rewrite the statement "No good cars are cheap" in the form "$\forall x$ if + $P(x)$ then $\neg Q(x)$." Indicate whether each of the following arguments + is valid or invalid, and justify your answers. + +a. + +$$ +\text{No good car is cheap.} \\ +\text{A Rimbaud is a good car.} \\ +\therefore \text{A Rimbaud is not cheap.} +$$ + +b. + +$$ +\text{No good car is cheap.} \\ +\text{A Simbaru is not cheap.} \\ +\therefore \text{A Simbaru is a good car.} +$$ + +c. + +$$ +\text{No good car is cheap.} \\ +\text{A VX Roadster is cheap.} \\ +\therefore \text{A VX Roadster is not good.} +$$ + +d. + +$$ +\text{No good car is cheap.} \\ +\text{An Omnex is not a good car.} \\ +\therefore \text{An Omnex is cheap.} +$$ + +20. + +a. Use a diagram to show that the following argument can have true premises and +a false conclusion. + +$$ +\text{All dogs are carnivorous.} \\ +\text{Aaron is not a dog.} \\ +\therefore \text{Aaron is not carnivorous.} +$$ + +b. What can you conclude about the validity or invalidity of the following +argument form? Explain how the result from part (a) leads to this conclusion. + +$$ +\forall x, \text{ if } P(x) \text{ then } Q(x) \\ +\neg P(a) \text{ for a particular } a \\ +\therefore \neg Q(a) +$$ + +Indicate whether the arguments in 21-27 are valid or invalid. Support your +answers by drawing diagrams. + +21. + +$$ +\text{All people are mice.} \\ +\text{All mice are mortal.} \\ +\therefore \text{All people are mortal.} +$$ + +22. + +$$ +\text{All discrete mathematics students can tell a valid argument from an invalid one.} \\ +\text{All thoughtful people can tell a valid argument from an invalid one.} \\ +\therefore \text{All discrete mathematics students are thoughtful.} +$$ + +23. + +$$ +\text{All teachers occasionally make mistakes} \\ +\text{No gods ever make mistakes.} \\ +\therefore \text{No teachers are gods.} +$$ + +24. + +$$ +\text{No vegetarians eat meat.} \\ +\text{All vegans are vegetarians.} \\ +\therefore \text{No vegans eat meat.} +$$ + +25. + +$$ +\text{No college cafeteria food is good.} \\ +\text{No good food is wasted.} \\ +\therefore \text{No college cafeteria food is wasted.} +$$ + +26. + +$$ +\text{All polynomial functions are differentiable.} \\ +\text{All differentiable functions are continuous.} \\ +\therefore \text{All polynomial functions are continuous.} +$$ + +27. + +[Adapted from Lewis Carrol.] + +$$ +\text{Nothing intelligible ever puzzles me.} \\ +\text{Logic puzzles me.} \\ +\therefore \text{Logic is unintelligible.} +$$ + +In exercises 28-32, reorder the premises in each of the arguments to show that +the conclusion follows as valid consequence from the premises. It may be helpful +to rewrite the statements in if-then form and replace some of them by their +contrapositives. Exercises 28-30 refer to the kinds of Tarski words discussed in +Examples 3.1.13 and 3.3.1. Exercises 31 and 32 are adapted from _Symbolic Logic_ +by Lewis Carroll. + +28. + + 1. Every object that is to the right of all the blue objects is above all the triangles. + + 2. If an object is a circle, then it is to the right of all the blue objects. + + 3. If an object is not a circle, then it is not gray. + +$\therefore$ All the gray objects are above all the triangles. + +29. + + 1. All the objects that are to the right of all the triangles are above all the circles. + + 2. If an object is not above all the black objects, then it is not a square. + + 3. All the objects that are above all the black objects are to the right of all the triangles. + +$\therefore$ All the squares are above all the circles. + +30. + + 1. If an object is above all the triangles, then it is above all the blue objects. + + 2. If an object is not above all the gray objects, then it is not a square. + + 3. Every black object is a square. + + 4. Every object that is above all the gray objects is above all the triangles. + +$\therefore$ If an object is black, then it is above all the blue objects. + +31. + + 1. I trust every animal that belongs to me. + + 2. Dogs gnaw bones. + + 3. I admit no animals into my study unless they will beg when told to do so. + + 4. All the animals in the yard are mine. + + 5. I admit every animal that I trust into my study. + + 6. The only animals that are really willing to beg when told to do so are dogs. + +$\therefore$ All the animals in the yard gnaw bones. + +32. + + 1. When I work a logic example without grumbling, you may be sure it is one I understand. + + 2. The arguments in these examples are not arranged in regular order like the ones I am used to. + + 3. No easy examples make my head ache. + + 4. I can't understand examples if the arguments are not arranged in regular order like the ones I am used to. + + 5. I never grumble at an example unless it gives me a headache. + +$\therefore$ These examples are not easy. + +In 33 and 34 a single conclusion follows when all the given premises are taken +into consideration, but it is difficult to see because the premises are jumbled +up. Reorder the premises to make it clear that a conclusion follows logically, +and state the valid conclusion that can be drawn. (It may be helpful to rewrite +some of the statements in if-then form and to replace some statements by their +contrapositives.) + +33. + + 1. No birds except ostriches are at least 9 feet tall. + + 2. There are no birds in this aviary that belong to anyone but me. + + 3. No ostrich lives on mince pies. + + 4. I have no birds less than 9 feet high. + +34. + + 1. All writers who understand human nature are clever. + + 2. No one is a true poet unless he can stir the human heart. + + 3. Shakespeare wrote Hamlet. + + 4. No writer who does not understand human nature can stir the human heart. + + 5. None but a true poet could have written Hamlet. + +35. Derive the validity of universal modus tollens from the validity of + universal instantiation and modus tollens. + +36. Derive the validity of universal form of part (a) of the elimination rule + from the validity of universal instantiation and the valid argument called + elimination in Section 2.3. diff --git a/chapter_3/notes.md b/chapter_3/notes.md index fb22402..8ea646d 100644 --- a/chapter_3/notes.md +++ b/chapter_3/notes.md @@ -163,8 +163,141 @@ $y$ in $E$ anyone might choose to challenge you with. --- +Page 160 + **Negations of Statements with Two Different Quantifiers** $\neg(\forall x \text{ in } d, \exists y \text{ in } E \text{ such that } P(x, y)) \equiv \exists x \text{ in } D \text{ such that } \forall y \text{ in } E, \neg P(x, y)$ $\neg(\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)) \equiv \forall x \text{ in } D, \exists y \text{ in } E \text{ such that } \neg P(x, y)$ + +--- + +Page 169 + +**Universal Instantiation** + +If a property is true of _everything_ in a set, then it is true of _any +particular_ thing in the set. + +--- + +Page 170 + +**Universal Modus Ponens** + +_Formal Version_ + +$$ +\forall x, \text{ if } P(x) \text{ then } Q(x) \\ +P(a) \text{ for a particular } a \\ +\therefore Q(a) +$$ + +_Informal Version_ + +$$ +\text{If } x \text{ makes } P(x) \text{true, then } x \text{ makes } Q(x) \text{ true.} \\ +a \text{ makes } P(x) \text{ true.} \\ +\therefore a \text{ makes } Q(x) \text{ true.} +$$ + +--- + +Page 172 + +**Universal Modus Tollens** + +_Formal Version_ + +$$ +\forall x, \text{ if } P(x) \text{ then } Q(x) \\ +\neg Q(a) \text{ for a particular } a \\ +\therefore \neg P(a) +$$ + +_Informal Version_ + +$$ +\text{If } x \text{ makes } P(x) \text{true, then } x \text{ makes } Q(x) \text{ true.} \\ +a \text{ does not make } Q(x) \text{ true.} \\ +\therefore a \text{ does not make } P(x) \text{ true.} +$$ + +--- + +Page 173 + +**Definition** + +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 **valid** if, and only if, its form is valid. It +is called _sound_ if, and only if, its form is valid and its premises are true. + +--- + +Page 176 + +**Converse Error (Quantified Form)** + +_Formal Version_ + +$$ +\forall x, \text{ if } P(x) \text{ then } Q(x) \\ +Q(a) \text{ for a particular } a \\ +\therefore \neg P(a) \text{ is an invalid conclusion} +$$ + +_Informal Version_ + +$$ +\text{If } x \text{ makes } P(x) \text{true, then } x \text{ makes } Q(x) \text{ true.} \\ +a \text{ makes } Q(x) \text{ true.} \\ +\therefore a \text{ makes } P(x) \text{ true. } \text{ is an invalid conclusion} +$$ + +--- + +Page 176 + +**Inverse Error (Quantified Form)** + +_Formal Version_ + +$$ +\forall x, \text{ if } P(x) \text{ then } Q(x) \\ +\neg P(a) \text{ for a particular } a \\ +\therefore \neg \neg Q(a) \text{ is an invalid conclusion} +$$ + +_Informal Version_ + +$$ +\text{If } x \text{ makes } P(x) \text{true, then } x \text{ makes } Q(x) \text{ true.} \\ +a \text{ does not make } P(x) \text{ true.} \\ +\therefore a \text{ does not make } Q(x) \text{ true. } \text{ is an invalid conclusion} +$$ + +--- + +Page 177 + +**Universal Transitivity** + +_Formal Version_ + +$$ +\forall x P(x) \to Q(x) \\ +\forall x Q(x) \to R(x) \\ +\therefore \forall x P(x) \to R(x) +$$ + +_Informal Version_ + +$$ +\text{Any } x \text{ that makes } P(x) \text{ true makes } Q(x) \text{ true.} \\ +\text{Any } x \text{ that makes } Q(x) \text{ true makes } R(x) \text{ true.} \\ +\therefore \text{Any } x \text{ that makes } P(x) \text{ true makes } R(x) \text{ true.} \\ +$$ diff --git a/chapter_3/test_yourself.md b/chapter_3/test_yourself.md index 05b9236..902dae9 100644 --- a/chapter_3/test_yourself.md +++ b/chapter_3/test_yourself.md @@ -112,3 +112,28 @@ c. may be true or may be false. c is the answer, it may be true or false depending on the nature of the property involving $x$, and $y$. In other words, it relies on what $P(x, y)$ states. + +--- + +**Test Yourself** + +Page 179 + +1. The rule of universal instantiation says that if some property is true for + _______ in a domain, then it is true for _______. + +2. If the first two premises of universal modus ponens are written as "If $x$ + makes $P(x)$ true, then $x$ makes $Q(x)$ true" and "For a particular value of + $a$ _______ , " then the conclusion can be written as "______. " + +3. If the first two premises of universal modus tollens are written as "If $x$ + makes $P(x)$ true, then $x$ makes $Q(x)$ true" and "For a particular value of + $a$ _______ ," then the conclusion can be written as " _______. " + +4. If the first two premises of universal transitivity are written as "Any $x$ + that makes $P(x)$ true makes $Q(x)$ true" and "Any $x$ that makes $Q(x)$ true + makes $R(x)$ true," then the conclusion can be written as "_______." + +5. Diagrams can be helpful in testing an argument for validity. However, if some + possible configurations of the premises are not drawn, a person could + conclude that an argument was _______ when it was actually _______.