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Chapter 1: Problem 5

What is the negation of each of these propositions? a) Mei has an MP3 player. b) There is no pollution in New Jersey. c) \(2+1=3\) . d) The summer in Maine is hot and sunny.

Short Answer

Expert verified
Negations: a) Mei does not have an MP3 player, b) There is pollution in New Jersey, c) \(2+1 e 3\), d) The summer in Maine is not hot and not sunny.

Step by step solution

01

Negating Proposition a

To negate the proposition 'Mei has an MP3 player,' simply state that Mei does not have an MP3 player.
02

Negating Proposition b

To negate the proposition 'There is no pollution in New Jersey,' state that there is pollution in New Jersey. Negating involves removing the 'no' from the statement.
03

Negating Proposition c

To negate the proposition '\( 2+1=3 \),' state that \( 2+1 e 3 \). You change the equals sign to a not equal sign.
04

Negating Proposition d

To negate the proposition 'The summer in Maine is hot and sunny,' state that the summer in Maine is not hot and not sunny. Negate each condition separately, and use 'and' to connect them.

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Key Concepts

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

Propositional Logic
Propositional logic, also known as statement logic, is the area of logic that deals with propositions and their connectives. A proposition is a statement that can either be true or false, unlike questions or commands. We use symbols to represent propositions
For example:
  • Let p represent 'Mei has an MP3 player'.
  • Let q represent 'There is no pollution in New Jersey'.
  • Let r represent '2+1=3'.
  • Let s represent 'The summer in Maine is hot and sunny'.
With these symbols, we can create more complex logical expressions by using various logical connectives like and (鈭�), or (鈭�), not (卢), implies (鈫�) and if and only if (鈫�). Understanding propositional logic is essential for constructing and analyzing logical arguments and statements.
Logical Negation
Negation is a basic operation in propositional logic. The negation of a proposition alters its truth value. If the original proposition is true, its negation is false, and vice versa. The notation for negation is 卢, which stands for 'not'. Here are examples using our previous propositions:
  • Negation of p: 卢p represents 'Mei does not have an MP3 player'.
  • Negation of q: 卢q represents 'There is pollution in New Jersey'.
  • Negation of r: 卢r represents '2+1 鈮� 3'.
  • Negation of s: 卢s represents 'The summer in Maine is not hot and sunny'.
When we negate a compound statement, we need to negate each part of the statement. For instance, to negate 's', we say 'The summer in Maine is not hot and not sunny'. Each part of the original statement has been negated.
Logical Statements
Logical statements are declarative sentences that can be either true or false. These statements are the foundation of propositional logic. There are some important properties and operations within logical statements:
  • **Conjunction (鈭�)**: The conjunction of A and B (A 鈭� B) is true if both A and B are true.
  • **Disjunction (鈭�)**: The disjunction of A and B (A 鈭� B) is true if at least one of A or B is true.
  • **Implication (鈫�)**: A implies B (A 鈫� B) is true either if A is false or B is true.
  • **Biconditional (鈫�)**: A if and only if B (A 鈫� B) is true if both A and B are either true or false.
From the exercise above, we work with negations. Understanding how to form logical statements and negations helps in simplifying and solving logical problems. Remember, logical negation changes the truth value of the statement: when a statement is true, its negation is false, and when a statement is false, its negation is true.

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Most popular questions from this chapter

The Logic Problem, taken from \(W F F^{\prime} N\) PROOF, The Game of Logic, has these two assumptions: 1\. "Logic is difficult or not many students like logic." 2\. "If mathematics is easy, then logic is not difficult." By translating these assumptions into statements involving propositional variables and logical connectives, de- termine whether each of the following are valid conclusions of these assumptions: a) That mathematics is not easy, if many students like logic. b) That not many students like logic, if mathematics is not easy. c) That mathematics is not easy or logic is difficult. d) That logic is not difficult or mathematics is not easy. e) That if not many students like logic, then either mathematics is not easy or logic is not difficult.

Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). a) \(\neg \forall x \forall y P(x, y) \quad\) b) \(\neg \forall y \exists x P(x, y)\) c) \(\neg \forall y \forall x(P(x, y) \vee Q(x, y))\) d) \(\neg(\exists x \exists y \neg P(x, y) \wedge \forall x \forall y Q(x, y))\) e) \(\quad \neg \forall x(\exists y \forall z P(x, y, z) \wedge \exists z \forall y P(x, y, z))\)

Suppose that the domain of the propositional function \(P(x)\) consists of the integers \(0,1,2,3,\) and \(4 .\) Write out each of these propositions using disjunctions, conjunctions, and negations. $$ \begin{array}{llll}{\text { a) }} & {\exists x P(x)} & {\text { b) } \forall x P(x)} & {\text { c) }} \quad {\exists x \neg P(x)} \\ {\text { d) }} & {\forall x \neg P(x)} & {\text { e) } \neg \exists x P(x)} & {\text { f) } \neg \forall x P(x)}\end{array} $$

Suppose that the domain of \(Q(x, y, z)\) consists of triples \(x, y, z,\) where \(x=0,1,\) or \(2, y=0\) or \(1,\) and \(z=0\) or \(1 .\) Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \forall y Q(0, y, 0)} & {\text { b) } \exists x Q(x, 1,1)} \\ {\text { c) } \exists z \neg Q(0,0, z)} & {\text { d) } \exists x \neg Q(x, 0,1)}\end{array} $$

Use quantifiers and logical connectives to express the fact that every linear polynomial (that is, polynomial of degree 1 ) with real coefficients and where the coefficient of \(x\) is nonzero, has exactly one real root.
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