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Chapter 1: Problem 38

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

Short Answer

Expert verified
Counterexamples are: a) x=0, x=1, b) x=\sqrt{2}, c) x=0

Step by step solution

01

Title - Understand the Statement

Interpret each of the given universally quantified statements one by one. Identify the domain and what the statement suggests about every element in the domain.
02

Title - Find Counterexample for Statement a

Statement a suggests that for every real number x, the equation \( x^2 eq x \) holds. Try to find a real number x where this is not true. Consider simple values such as 0 and 1. \ For x = 0: \( 0^2 = 0 \) which does not satisfy \( x^2 eq x \). So, x = 0 is a counterexample. \ For x = 1: \( 1^2 = 1 \) which also does not satisfy \( x^2 eq x \). So, x = 1 is another counterexample.
03

Title - Find Counterexample for Statement b

Statement b suggests that for every real number x, \( x^2 eq 2 \) holds. Try to find a real number x where this statement does not hold. \ The value \( x = \sqrt{2} \) gives us \( (\sqrt{2})^2 = 2 \), which does not satisfy \( x^2 eq 2 \). So, \( x = \sqrt{2} \) is a counterexample.
04

Title - Find Counterexample for Statement c

Statement c suggests that for every real number x, \( |x| > 0 \) holds. Try to find a real number x where this is not true. \ For x = 0: \( |0| = 0 \) which does not satisfy \( |x| > 0 \). So, x = 0 is a counterexample.

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Key Concepts

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

Counterexample in Mathematics
In mathematics, a counterexample is an example that disproves a universally quantified statement. This means that if a statement says 'for all' elements in a domain, doing something, a counterexample shows that at least one element in that domain does not comply with the statement.

For instance, let's say we have the statement: 'For all real numbers x, x^2 is not equal to x.' If we find any real number x for which x^2 equals x, we have found a counterexample that disproves the statement.

Counterexamples are extremely useful in understanding the limits of a statement and are a key tool in the mathematician's toolkit. They help keep our mathematical claims precise and accurate.
Universally Quantified Statements
A universally quantified statement in mathematics declares that a property or condition is true for all members of a specified domain. These statements often use the universal quantifier symbol \( \forall \), which means 'for all' or 'for every'.

For example, the statement \( \forall x \left(x^2 eq x \right) \) means 'For all x, x squared is not equal to x'. To fully understand and potentially disprove such statements, you need to evaluate them within their given domains, such as all real numbers.

Universally quantified statements are critical in mathematical theorems and proofs. However, just one counterexample is enough to disprove them.
Real Numbers Domain
In the context of our exercise, the domain of all variables consists of all real numbers. Real numbers include all the numbers that can be found on the number line. This includes:
  • Positive numbers
  • Negative numbers
  • Zero
  • Irrational numbers (such as \( \sqrt{2} \) and \( \pi \))
  • Rational numbers (such as fractions and integers)
Understanding the domain is crucial when dealing with universally quantified statements because it tells us where we should be looking for counterexamples. For instance, if the domain was only positive integers, our focus would change.

The statement \( |x| > 0 \) in the domain of real numbers is interesting. While it might hold for all positive or negative numbers, it fails at x = 0 since \( |0| = 0 \).
Understanding these different elements helps us find accurate counterexamples. Ensuring that you correctly consider the domain can make a significant difference in proving or disproving universally quantified statements.

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Most popular questions from this chapter

Show that each of these statements can be used to ex- press the fact that there is a unique element \(x\) such that \(P(x)\) is true. [Note that we can also write this statement as \(\exists ! x P(x) . ]\) a) \(\exists x \forall y(P(y) \leftrightarrow x=y)\) b) \(\exists x P(x) \wedge \forall x \forall y(P(x) \wedge P(y) \rightarrow x=y)\) c) \(\exists x(P(x) \wedge \forall y(P(y) \rightarrow x=y))\)

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) \(\quad \neg \forall x(\exists y \forall z P(x, y, z) \wedge \exists z \forall y P(x, y, z))\)

Show that the argument form with premises \(p_{1}, p_{2}, \ldots, p_{n}\) and conclusion \(q \rightarrow r\) is valid if the argument form with premises \(p_{1}, p_{2}, \ldots, p_{n}, q,\) and conclusion \(r\) is valid. Show that the argument form with premises \(p_{1}, p_{2}, \ldots, p_{n}\) and conclusion \(q \rightarrow r\) is valid if the argument form with premises \(p_{1}, p_{2}, \ldots, p_{n}, q,\) and conclusion \(r\) is valid.

Write out \(\exists ! x P(x),\) where the domain consists of the integers \(1,2,\) and \(3,\) in terms of negations, conjunctions, and disjunctions.

Prove or disprove that if you have an 8 -gallon jug of water and two empty jugs with capacities of 5 gallons and 3 gallons, respectively, then you can measure 4 gallons by successively pouring some of or all of the water in a jug into another jug.
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