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Chapter 1: Problem 7

Express these system specifications using the propositions \(p :\) "The message is scanned for viruses" and \(q :\) "The message was sent from an unknown system" together with logical connectives (including negations). a) "The message is scanned for viruses whenever the message was sent from an unknown system." b) 鈥淭he message was sent from an unknown system but it was not scanned for viruses.鈥� c) 鈥淚t is necessary to scan the message for viruses when- ever it was sent from an unknown system.鈥� d) 鈥淲hen a message is not sent from an unknown system it is not scanned for viruses.鈥�

Short Answer

Expert verified
a) \( q \rightarrow p \) b) \( q \land eg p \) c) \( q \rightarrow p \) d) \( eg q \rightarrow eg p \)

Step by step solution

01

Title - Define the propositions

Let the propositions be defined as follows: \( p : \text{The message is scanned for viruses} \)\( q : \text{The message was sent from an unknown system} \)
02

- Express specification (a)

The phrase 'The message is scanned for viruses whenever the message was sent from an unknown system' can be written as: \( q \rightarrow p \)This means that if the message was sent from an unknown system (\( q \)), then it is scanned for viruses (\( p \)).
03

- Express specification (b)

The phrase 'The message was sent from an unknown system but it was not scanned for viruses' can be written as: \( q \land eg p \)This means that the message was sent from an unknown system (\( q \)) and it was not scanned for viruses (\( eg p \)).
04

- Express specification (c)

The phrase 'It is necessary to scan the message for viruses whenever it was sent from an unknown system' can be written similarly to specification (a): \( q \rightarrow p \)This is another way of saying that it is required to scan the message for viruses (\( p \)) if it was sent from an unknown system (\( q \)).
05

- Express specification (d)

The phrase 'When a message is not sent from an unknown system it is not scanned for viruses' can be written as: \( eg q \rightarrow eg p \)This means that if a message is not sent from an unknown system (\( eg q \)), then it is not scanned for viruses (\( eg p \)).

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Key Concepts

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

Logical Connectives
In propositional logic, logical connectives are symbols or words used to connect two or more propositions. These connectives help in forming complex logical expressions and thus enable us to express various logical relations and conditions.
Some common logical connectives are:
  • AND ( \land ): This connects two propositions and yields true only if both propositions are true. For example, (\(p \land q\)) is true if both \(p\) and \(q\) are true.
  • OR ( \lor ): This connective yields true if at least one of the propositions is true. (\(p \lor q\)) is true if either \(p\) or \(q\) is true.
  • NOT ( eg ): This negates the proposition. For example, (\( eg p\)) is true if \(p\) is false.
  • IMPLIES ( \rightarrow ): This represents implication. (\(p \rightarrow q\)) means if \(p\) is true, then \(q\) must also be true.
Understanding these connectives is crucial for expressing complex situations and statements in logic.
Propositional Equivalence
Propositional equivalence occurs when two propositions always have the same truth value. In other words, they are true in exactly the same scenarios and false in the same scenarios. This concept is useful for simplifying logical expressions.
For instance:
  • The statement (\( p \land q\)) is equivalent to saying (\( q \land p\)). Both mean that \(p\) and \(q\) are true, regardless of their order.
  • The implications (\( q \rightarrow p\)) and (\( eg p \rightarrow eg q\)) are logically equivalent. This can be seen in the exercise: 'When a message is not sent from an unknown system, it is not scanned for viruses' (\( eg q \rightarrow eg p\)) is logically equivalent to 'The message is scanned for viruses whenever it is sent from an unknown system' (\( q \rightarrow p\)).
Recognizing such equivalences simplifies reasoning about logical statements and deriving results from given logical conditions.
Implication
Implication in logic (also called 'conditional') is an important concept represented by the connective (\( \rightarrow \)). An implication (\( p \rightarrow q\)) means: 'if \(p\) is true, then \(q\) must be true'.
Key points about implications:
  • It can be true even if both preceding proposition \(p\) and the resultant \(q\) are false.
  • It is only false if \(p\) is true and \(q\) is false.
  • In the exercise, 'The message is scanned for viruses whenever it was sent from an unknown system' is written as (\( q \rightarrow p\)). Here, if the message was sent from an unknown system (\(q\)), then it must be scanned for viruses (\(p\)).
Understanding implication helps determine what conditions must be true for specific outcomes and forms the basis of logical deductions.

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Most popular questions from this chapter

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�

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