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Chapter 1: Problem 12

Let \(p\) and \(q\) be the propositions "The election is decided" and "The votes have been counted, " respectively. Express each of these compound propositions an English sentence. a) \(\neg p\) b) \(p \vee q\) c) \(\neg p \wedge q\) d) \(q \rightarrow p\) e) \(\neg q \rightarrow \neg p\) f) \(\neg p \rightarrow \neg q\) g) \(p \leftrightarrow q\) h) \(\neg q \vee(\neg p \wedge q)\)

Short Answer

Expert verified
a) The election is not decided. b) The election is decided or the votes have been counted. c) The election is not decided and the votes have been counted. d) If the votes have been counted, then the election is decided. e) If the votes have not been counted, then the election is not decided. f) If the election is not decided, then the votes have not been counted. g) The election is decided if and only if the votes have been counted. h) The votes have not been counted or (the election is not decided and the votes have been counted).

Step by step solution

01

- Understanding Basic Propositions

Identify the basic propositions from the given statements: p: The election is decided q: The votes have been counted
02

- Negation of p

a) \(eg p\): The negation of 'The election is decided' is 'The election is not decided'.
03

- Disjunction of p and q

b) \(p \vee q\): This represents the disjunction 'The election is decided or the votes have been counted'.
04

- Conjunction of Negation of p and q

c) \(eg p \wedge q\): This represents the conjunction 'The election is not decided and the votes have been counted'.
05

- Conditional q implies p

d) \(q \rightarrow p\): This represents the conditional 'If the votes have been counted, then the election is decided'.
06

- Conditional Negation of q implies Negation of p

e) \(eg q \rightarrow eg p\): This represents the conditional 'If the votes have not been counted, then the election is not decided'.
07

- Conditional Negation of p implies Negation of q

f) \(eg p \rightarrow eg q\): This represents the conditional 'If the election is not decided, then the votes have not been counted'.
08

- Biconditional p if and only if q

g) \(p \leftrightarrow q\): This represents the biconditional 'The election is decided if and only if the votes have been counted'.
09

- Complex Expression

h) \(eg q \vee (eg p \wedge q)\): This represents the disjunction 'The votes have not been counted or (the election is not decided and the votes have been counted)'.

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Key Concepts

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

Logic Expressions
In propositional logic, a logic expression is a combination of propositions using logical connectives. Propositions are simple statements that can either be true or false. Logical expressions combine these propositions to create more complex statements. For example, if we have propositions like 'The election is decided' ( moveso), and 'The votes have been counted' (P moveso), we combine them with connectives like 'and', 'or', and 'not' to make expressions like P 鈭� Q. By understanding these connectives, we can interpret complex logical statements.
Negations
Negations are used to express the opposite of a given proposition. If a proposition P is 'The election is decided', then its negation P would be 'The election is not decided'. The symbol for negation is \(eg\).
A practical example from the exercise is: \(eg p\), which translates to, 'The election is not decided'.
By understanding negations, we can better analyze what it means for a proposition to be false.
Conjunctions
A conjunction is a compound statement formed using 'and'. It is true only if both propositions are true.
The symbol for conjunction is \(\wedge\). For example, \(eg p \wedge q\) means, 'The election is not decided and the votes have been counted.'
Using conjunctions helps you understand how multiple conditions must be satisfied for the overall statement to be true.
Disjunctions
Disjunctions connect propositions using 'or'. The compound statement is true if at least one of the propositions is true.
The symbol for disjunction is \(\vee\). For instance, \(p \vee q\) means, 'The election is decided or the votes have been counted.'
By learning about disjunctions, you can see how one true proposition can make the whole statement true even if the other is false.
Conditionals
Conditionals create an 'if-then' relationship between propositions. If the first proposition is true, it implies the second one is true. The symbol for conditionals is \(\rightarrow\).
For instance, \(q \rightarrow p\) means, 'If the votes have been counted, then the election is decided.'
Understanding conditionals is crucial for grasping cause-and-effect relationships in logic.
Biconditionals
Biconditionals state that two propositions are true or false together. The symbol is \(\leftrightarrow\).
For example, \(p \leftrightarrow q\) translates to 'The election is decided if and only if the votes have been counted.'
Getting familiar with biconditionals helps you recognize when two conditions always happen together.

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Most popular questions from this chapter

Find all squares, if they exist, on an \(8 \times 8\) checkerboard such that the board obtained by removing one of these squares can be tiled using straight triominoes. [Hint: First use arguments based on coloring and rotations to eliminate as many squares as possible from consideration. \(]\)

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Let M(x, y) be 鈥渪 has sent y an e-mail message鈥� and T(x, y) be 鈥渪 has telephoned y,鈥� where the domain consists of all students in your class. Use quantifiers to express each of these statements. (Assume that all e-mail messages that were sent are received, which is not the way things often work.) a) Chou has never sent an e-mail message to Koko. b) Arlene has never sent an e-mail message to or tele- phoned Sarah. c) Jose has never received an e-mail message from Deborah. d) Every student in your class has sent an e-mail mes- sage to Ken. e) No one in your class has telephoned Nina. f ) Everyone in your class has either telephoned Avi or sent him an e-mail message. g) There is a student in your class who has sent every- one else in your class an e-mail message. h) There is someone in your class who has either sent an e-mail message or telephoned everyone else in your class. i) There are two different students in your class who have sent each other e-mail messages. j) There is a student who has sent himself or herself an e-mail message. k) There is a student in your class who has not received an e-mail message from anyone else in the class and who has not been called by any other student in the class. l) Every student in the class has either received an email message or received a telephone call from another student in the class. m) There are at least two students in your class such that one student has sent the other e-mail and the second student has telephoned the first student. n) There are two different students in your class who between them have sent an e-mail message to or telephoned everyone else in the class.
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