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Chapter 2: Problem 7

For each statement, (i) represent it as a formula, (ii) find the negation (in simplest form) of this formula, and (iii) express the negation in words. (a) For all real numbers \(x\) and \(y, x+y=y+x\). (b) For every positive real number \(x\) there exists a real number \(y\) such that \(y^{2}=x\). (c) There exists a real number \(y\) such that, for every integer \(x, 2 x^{2}+1>x^{2} y\).

Short Answer

Expert verified
(a) "There exist \(x\) and \(y\) such that \(x+y \neq y+x\)." (b) "There exists a positive \(x\) such that \(y^2 \neq x\) for all \(y\)." (c) "For every \(y\), \(2x^2+1 \leq x^2y\) for some \(x\)."

Step by step solution

01

Translate Statement (a) into a Formula

The statement "For all real numbers \(x\) and \(y, x+y=y+x\)" is universally quantified. In logical notation, this can be represented as:\[ \forall x \forall y (x+y = y+x) \]
02

Find the Negation of Statement (a)

To negate the statement \( \forall x \forall y (x+y = y+x) \), use the rule for negating universally quantified statements:\[ \exists x \exists y eg (x+y = y+x) \] which simplifies to\[ \exists x \exists y (x+y eq y+x) \]
03

Express the Negation of Statement (a) in Words

The negation \( \exists x \exists y (x+y eq y+x) \) translates to: "There exist real numbers \(x\) and \(y\) such that \(x+y\) is not equal to \(y+x\)."
04

Translate Statement (b) into a Formula

The statement "For every positive real number \(x\) there exists a real number \(y\) such that \(y^{2}=x\)" is a combination of universal and existential quantifiers. It can be represented as:\[ \forall x > 0, \exists y (y^2 = x) \]
05

Find the Negation of Statement (b)

To negate \( \forall x > 0, \exists y (y^2 = x) \), use the rule for negating statements with mixed quantifiers:\[ \exists x > 0, \forall y eg (y^2 = x) \] which simplifies to\[ \exists x > 0, \forall y (y^2 eq x) \]
06

Express the Negation of Statement (b) in Words

The negation \( \exists x > 0, \forall y (y^2 eq x) \) translates to: "There exists a positive real number \(x\) such that for all real numbers \(y\), \(y^2\) is not equal to \(x\)."
07

Translate Statement (c) into a Formula

The statement "There exists a real number \(y\) such that, for every integer \(x, 2x^{2}+1>x^{2} y\)" can be expressed as:\[ \exists y \forall x \in \mathbb{Z} (2x^{2}+1 > x^{2} y) \]
08

Find the Negation of Statement (c)

Negating \( \exists y \forall x \in \mathbb{Z} (2x^{2}+1 > x^{2} y) \) results in:\[ \forall y \exists x \in \mathbb{Z} eg (2x^{2}+1 > x^{2} y) \] which simplifies to\[ \forall y \exists x \in \mathbb{Z} (2x^{2}+1 \leq x^{2} y) \]
09

Express the Negation of Statement (c) in Words

The negation \( \forall y \exists x \in \mathbb{Z} (2x^{2}+1 \leq x^{2} y) \) translates to: "For every real number \(y\), there exists an integer \(x\) such that \(2x^{2}+1\) is less than or equal to \(x^{2} y\)."

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Key Concepts

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

Understanding Quantifiers
Quantifiers are symbols used in logic to express the generality or existence of elements in a statement. There are two primary types of quantifiers: universal and existential.
  • **Universal Quantifier (\(\forall\))**: This is used to indicate that a statement applies to all members of a set. For example, \(\forall x\) means "for all \(x\)." In the exercise, statements (a) and (b) use this quantifier to express a condition applicable to all elements, such as all real numbers or positive real numbers.

  • **Existential Quantifier (\(\exists\))**: This denotes the existence of at least one element for which the statement holds true. The symbol \(\exists\) indicates there exists some element, like a real number \(y\), satisfying a condition. Statement (c) employs this quantifier to indicate the existence of a real number with a specific property.

Understanding how to use and negate these quantifiers is crucial in logic, as this helps us manipulate and comprehend logical statements in mathematical reasoning.
Logical Statements in Mathematics
Logical statements are declarative sentences that are either true or false. In mathematics, these are used to build assertions and proofs.
  • **Structure**: Logical statements consist of subjects, predicates, and connectives. They describe relationships between elements or properties within a specific domain, such as real numbers or integers.

  • **Negation**: Negating a logical statement involves changing its truth value. This is often done by altering the quantifiers and predicates in a statement. For example, the negation of a universally quantified statement turns it into an existentially quantified statement with the opposite predicate.

Negating logical statements is crucial for developing a deeper understanding of their meaning and implications, particularly in mathematical contexts.
Real Numbers in Logical Statements
Real numbers provide a continuous and unbroken set of numbers used in mathematics. They can be positive, negative, or zero, occurring in logical statements to express conditions and relationships.
  • **Applications**: Logical statements often include real numbers to describe equations or inequalities, such as \(x + y = y + x\), which holds for all real numbers \(x\) and \(y\). This showcases properties like commutativity or existence of squares, as seen in other exercises.

  • **Negation Implications**: When negating statements with real numbers, it's important to consider changing equality to inequality or vice versa, as seen in the negation of the exercise components.

Real numbers play a vital role in expressing and proving logical relationships within mathematical domains.
Understanding Integers in Logic
Integers are whole numbers that include negative numbers, zero, and positive numbers. In logical context, they often appear when expressing conditions or sets, like the count of elements.
  • **Role in Logic**: Integers are typically part of logical statements involving discrete quantities or conditions. In statement (c), integers are used to assess relationships among other terms in inequalities.

  • **Handling Quantification**: When working with integers, it's essential to understand how they affect logical quantifiers. For example, changing the quantifier from existence to universality impacts the relationship expressed by the integers.

Integers are a fundamental component of logical statements used in mathematics to showcase relationships between countable elements.

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Most popular questions from this chapter

Simplify the following formulas: (a) \(p \wedge(p \wedge q)\) (b) \(\overline{\bar{p} \vee q}\) (c) \(\overline{p \Rightarrow \bar{q}}\)

Determine whether formulas \(u\) and \(v\) are logically equivalent (you may use truth tables or properties of logical equivalences). (a) \(u:(p \Rightarrow q) \wedge(p \Rightarrow \bar{q}) \quad v: \bar{p}\) (b) \(u: p \Rightarrow q\) \(v: q \Rightarrow p\) (c) \(u: p \Leftrightarrow q\) \(v: q \Leftrightarrow p\) (d) \(u:(p \Rightarrow q) \Rightarrow r\) \(v: p \Rightarrow(q \Rightarrow r)\)

Determine whether these statements are true or false: (a) There is a rational number \(x\) such that \(x^{2} \leq 0\). (b) There exists a number \(x\) such that for every real number \(y, x y=0\). (c) For all \(x \in \mathbb{Z},\) either \(x\) is even, or \(x\) is odd. (d) There exists a unique number \(x\) such that \(x^{2}=1\).

Indicate which of the following are propositions (assume that \(x\) and \(y\) are real numbers). (a) The integer 36 is even. (b) Is the integer \(3^{15}-8\) even? (c) The product of 3 and 4 is 11 . (d) The sum of \(x\) and \(y\) is 12 . (e) If \(x>2,\) then \(x^{2} \geq 3\). (f) \(5^{2}-5+3\).

In volleyball it is important to know which team is serving, because a team scores a point only if that team is serving and wins a volley. If the serving team loses the volley, then the other team gets to serve. Thus, to keep score in a volleyball game between teams \(A\) and \(B,\) it may be useful to define propositional variables \(p\) and \(q,\) where \(p\) is true if team \(A\) is serving (hence false if team \(B\) is serving); and \(q\) is true if team \(A\) wins the current volley (hence false if team \(B\) wins it). (a) Give a formula that is true if team \(A\) scores a point and is false otherwise. (b) Give a formula that is true if team \(B\) scores a point and is false otherwise. (c) Give a formula that is true if the serving team loses the current volley and is false otherwise. (d) Give a formula whose truth value determines whether the serving team will serve again.
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