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Chapter 1: Problem 44

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Express each of these system specifications using predicates, quantifiers, and logical connectives. a) Every user has access to an electronic mailbox. b) The system mailbox can be accessed by everyone in the group if the file system is locked. c) The firewall is in a diagnostic state only if the proxy server is in a diagnostic state. d) At least one router is functioning normally if the throughput is between 100 kbps and 500 kbps and the proxy server is not in diagnostic mode.

Short Answer

Expert verified
a) \(\forall x (U(x) \rightarrow M(x))\), b) \((S \rightarrow \forall x (G(x) \rightarrow A(x)))\), c) \((F \rightarrow P)\), d) \((T \land \eg P \rightarrow \exists x (R(x) \land N(x)))\)

Step by step solution

01

Identify Predicates and Quantifiers

First, identify the predicates and quantifiers needed for each statement. Let鈥檚 define the necessary predicates: - Let U(x) be 'x is a user' - Let M(x) be 'x has access to an electronic mailbox' - Let S be 'the file system is locked' - Let G(x) be 'x is in the group' - Let A(x) be 'x can access the system mailbox' - Let F be 'the firewall is in a diagnostic state' - Let P be 'the proxy server is in a diagnostic state' - Let R(x) be 'x is a router' - Let N(x) be 'x is functioning normally' - Let T be 'the throughput is between 100 kbps and 500 kbps'
02

Translate Statement a

The first statement is 'Every user has access to an electronic mailbox.' We use universal quantification here. The logical expression will be: \(\forall x (U(x) \rightarrow M(x))\)
03

Translate Statement b

The second statement is 'The system mailbox can be accessed by everyone in the group if the file system is locked.' We use implication and universal quantification. The logical expression will be: \((S \rightarrow \forall x (G(x) \rightarrow A(x)))\)
04

Translate Statement c

The third statement is 'The firewall is in a diagnostic state only if the proxy server is in a diagnostic state.' This is a conditional statement. The logical expression will be: \((F \rightarrow P)\)
05

Translate Statement d

The fourth statement is 'At least one router is functioning normally if the throughput is between 100 kbps and 500 kbps and the proxy server is not in diagnostic mode.' We use existential quantification and conjunction. The logical expression will be: \((T \land \eg P \rightarrow \exists x (R(x) \land N(x)))\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantifiers
Quantifiers are symbols used in logic to express the quantity of specimens to which a predicate applies. The two main types are:

  • **Universal quantifier** (\( \forall \)): Indicates that a predicate applies to all elements in a domain. E.g., \( \forall x (U(x) \rightarrow M(x)) \) means 'For all x, if x is a user, then x has access to an electronic mailbox.'
  • **Existential quantifier** (\( \thereexists \)): States that there exists at least one element in the domain for which the predicate holds. E.g., \( \thereexists x (R(x) \rightarrow N(x)) \) means 'There exists at least one x such that if x is a router, then x is functioning normally.'
Logical Expressions
Logical expressions are statements formulated using logical operators and predicates. They precisely describe conditions or relationships. Key elements are:

  • **Predicates**: Functions that return true or false for given inputs (e.g., \( U(x) \) for 'x is a user').
  • **Quantifiers**: Indicate the scope (universal or existential) as discussed above.
  • **Connectives**: Logical operators (like and, or, not) to combine predicates.
For example, the expression \( (S \rightarrow \forall x (G(x) \rightarrow A(x))) \) can be broken down as: 'If the system is locked, then for all x, if x is in the group, then x can access the system mailbox.'
Predicates
Predicates are statements concerning properties or relationships among variables. They can be true or false depending on their arguments. For instance:

  • \( U(x) \) could indicate 'x is a user.'
  • \( M(x) \) might mean 'x has access to an electronic mailbox.'
Combining predicates leads to more complex expressions. For instance, \( U(x) \rightarrow M(x) \) means 'If x is a user, then x has access to an electronic mailbox,' creating logical relationships among variables.
Logical Connectives
Logical connectives are used to combine predicates or logical statements. The primary connectives include:

  • **And ( \( \land \) )**: Both statements must be true (e.g., \( A \land B \) means both A and B are true).
  • **Or ( \( \lor \) )**: At least one statement must be true (e.g., \( A \or B \) means either A or B or both are true).
  • **Not ( \( \eg \) )**: Reverses the truth value (e.g., \( \eg A \) means A is not true).
  • **Implication ( \( \rightarrow \) )**: If the first statement is true, then the second must be true (e.g., \( A \rightarrow B \) means if A is true, B must also be true).
Consider the expression \( (T \land \eg P \rightarrow \exists x (R(x) \land N(x))) \), which can be read as 'If the throughput is between 100 kbps and 500 kbps and the proxy server is not in diagnostic mode, then there exists at least one router that is functioning normally.'

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Most popular questions from this chapter

Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.

For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. a) 鈥淚f I take the day off, it either rains or snows.鈥� 鈥淚 took Tuesday off or I took Thursday off.鈥� 鈥淚t was sunny on Tuesday.鈥� 鈥淚t did not snow on Thursday.鈥� b) 鈥淚f I eat spicy foods, then I have strange dreams.鈥� 鈥淚 have strange dreams if there is thunder while I sleep.鈥� 鈥淚 did not have strange dreams.鈥� c) 鈥淚 am either clever or lucky.鈥� 鈥淚 am not lucky.鈥� 鈥淚f I am lucky, then I will win the lottery.鈥� d) 鈥淓very computer science major has a personal computer.鈥� 鈥淩alph does not have a personal computer.鈥� 鈥淎nn has a personal computer.鈥� e) 鈥淲hat is good for corporations is good for the United States.鈥� 鈥淲hat is good for the United States is good for you.鈥� 鈥淲hat is good for corporations is for you to buy lots of stuff.鈥� f ) 鈥淎ll rodents gnaw their food.鈥� 鈥淢ice are rodents.鈥� 鈥淩abbits do not gnaw their food.鈥� 鈥淏ats are not ro- dents.鈥�

Determine whether each of these arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what logical error occurs? a) If \(n\) is a real number such that \(n>1,\) then \(n^{2}>1\) Suppose that \(n^{2}>1 .\) Then \(n>1\) b) If \(n\) is a real number with \(n>3,\) then \(n^{2}>9\) . Suppose that \(n^{2} \leq 9 .\) Then \(n \leq 3\) . c) If \(n\) is a real number with \(n>2,\) then \(n^{2}>4\) . Suppose that \(n \leq 2 .\) Then \(n^{2} \leq 4 .\)

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all real numbers. $$ \begin{array}{ll}{\text { a) } \forall x\left(x^{2} \neq x\right)} & {\text { b) } \forall x\left(x^{2} \neq 2\right)} \\ {\text { c) } \forall x(|x|>0)} \end{array} $$

Suppose that five ones and four zeros are arranged around a circle. Between any two equal bits you insert a 0 and between any two unequal bits you insert a 1 to produce nine new bits. Then you erase the nine original bits. Show that when you iterate this procedure, you can never get nine zeros. [Hint: Work backward, assuming that you did end up with nine zeros.]
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