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Chapter 2: Problem 9

9\. Let \(p(x), q(x)\), and \(r(x)\) be the following open statements. $$ \begin{array}{ll} p(x): & x^{2}-7 x+10=0 \\ q(x): & x^{2}-2 x-3=0 \\ r(x): & x<0 \end{array} $$ a) Determine the truth or falsity of the following statements, where the universe is all integers. If a statement is false, provide a counterexample or explanation. i) \(\forall x[p(x) \rightarrow \neg r(x)]\) ii) \(\forall x[q(x) \rightarrow r(x)]\) iii) \(\exists x[q(x) \rightarrow r(x)]\) iv) \(\exists x[p(x) \rightarrow r(x)]\) b) Find the answers to part (a) when the universe consists of all positive integers. c) Find the answers to part (a) when the universe contains only the integers 2 and 5 .

Short Answer

Expert verified
Part a) i) True ii) False iii) True iv) False Part b) i), ii), iii), and iv) are all False Part c) i) True ii), iii), and iv) are all False

Step by step solution

01

Understand the Open Statements

The exercise contains three open statements. \(p(x)\) and \(q(x)\) both represent quadratic equations, and will be true when their solutions in \(x\) are real numbers. Statement \(r(x)\) says that \(x\) is less than zero: this will be true when \(x\) is a negative number.
02

Analyze the Logical Statements

Each of the logical statements in the exercise forms a logical implication between one of the open statements. The implication \(P \rightarrow Q\) is false only when \(P\) is true and \(Q\) is false. Otherwise, it is true.
03

Solve the Quadratic Equations

By solving the quadratic equations of \(p(x)\) and \(q(x)\), we get the roots as \(x = 2, 5\) for \(p(x)\) and \(x = -1, 3\) for \(q(x)\). These are the integer values of \(x\) for which \(p(x)\) and \(q(x)\) are true.
04

Determine the Truth Values of the Statements (Universe: All Integers)

i) For confirmation: the statement means: For all \(x\), if \(x\) is a solution of \(p(x)\), then \(x\) is not negative. As both solutions of \(p(x)\) are positive, the statement is true. ii) This statement: For all \(x\), if \(x\) is a solution of \(q(x)\), then \(x\) is negative. As one of the solutions of \(q(x)\) is positive, the statement is false. iii) Statement: There exists an \(x\) such that if \(x\) is a solution of \(q(x)\), then \(x\) is negative. As one of the solutions of \(q(x)\) is negative, the statement is true. iv) Statement: There exists an \(x\) such that if \(x\) is a solution of \(p(x)\), then \(x\) is negative. As all solutions of \(p(x)\) are positive, the statement is false.
05

Determine the Truth Values of the Statements (Universe: All Positive Integers)

Here, all statements involving \(r(x)\) will be false as \(r(x)\) itself is always false in the universe of all positive integers.
06

Determine the Truth Values of the Statements (Universe: The Integers 2 and 5)

In this universe, \(r(x)\) is always false. i) The statement is true again because we have for all \(x\), if \(x\) is a solution of \(p(x)\), then \(x\) is not negative. ii), iii), and iv) are all false as the implications have all true antecedents (\(q(x)\) and \(p(x)\) for 2 and 5 are true) and false consequents (\(r(x)\) for 2 and 5 is false).

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Key Concepts

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

Mathematical Logic
Mathematical logic is a foundational part of mathematics that focuses on the use of formal logical systems to represent and reason about propositions and their relationships. It involves understanding how statements can be formed, combined, and evaluated in terms of truth and falsehood. This is particularly useful in areas like computer science or mathematics where precise reasoning is crucial.

In propositional logic, we often work with statements (propositions) that can either be true or false. These statements can be simple or as complex as needed, constructed using logical connectives such as "and", "or", "not", and "if...then". When dealing with open statements, like the ones in the exercise, mathematical logic allows us to evaluate these statements under different conditions or universes, helping us understand their behavior more thoroughly. Logical implications, for instance, are a critical tool in mathematical reasoning. An implication "if P then Q" is shown to be false only when P is true, yet Q is false.

Understanding and using these tools helps us solve complex problems by breaking them into smaller, logical parts, as seen in the given exercise, where we decide the truth value of logical statements within specific universes, such as all integers.
Open Statements
Open statements, in mathematical logic, are expressions that consist of one or more variables, which may turn into true or false propositions depending on the values given to these variables. These are essential when considering logical problems, as they allow us to introduce variables that can be manipulated for different situations.

For example, the open statements in the exercise like \(p(x): x^2 - 7x + 10 = 0\) depend on the value of \(x\). If \(x\) is replaced by one of its roots found by solving the quadratic equation, such as 2 or 5, then \(p(x)\) becomes a true statement because these values satisfy the equation. Conversely, if other values are used where the equation doesn't hold, the status shifts to false.

The same craftsmanship applies to \(q(x)\) and \(r(x)\). A logical analysis involves evaluating these statements within different "universes" or sets of possible values for \(x\). These universes are critical for evaluating logical propositions in diverse contexts, as seen when testing the truth of implications across different sets like positive integers, or just the integers 2 and 5.
Truth Values
Truth values in logic essentially define whether propositions are true or false. These values are fundamental when analyzing logical statements, particularly in determining the validity of expressions like implications, conjunctions, and disjunctions.

In the step-by-step solution, the concept of truth value plays a pivotal role. For example, the implication \( \forall x[p(x) \rightarrow eg r(x)] \) was analyzed to be true, as for all \(x\), if \(p(x)\) was true, \(x\) would not be negative, aligning with the fact that both solutions of \( p(x)\) were positive.

In contrast, for \( \forall x[q(x) \rightarrow r(x)] \), the statement is false because one root satisfies \( q(x)\), being negative, but the other does not. Evaluating each proposition in specific universes affects their truth values. For positive integers, \( r(x)\) is always false, influencing the truth values of combined statements in such scenarios.
  • All true implications generate true statements unless the consequent is false.
  • The negation flip the truth value.
  • Existentially quantified statements are true if there's at least one instance where the statement is true.
Understanding truth values offers a framework for logically assessing statements' validity across varying conditions.

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Most popular questions from this chapter

11\. Let \(p, q\), and \(r\) denote primitive statements. Find a form of the contrapositive of \(p \rightarrow(q \rightarrow r)\) with (a) only one occurrence of the connective \(\rightarrow\); (b) no occurrences of the connective \(\rightarrow\).

7\. For the universe of all integers, let \(p(x), q(x), r(x), s(x)\), nd \(t(x)\) be the following open statements. \(s(x) ; \quad x\) is (exactly) divisible by 4 \(t(x): \quad x\) is (exactly) divisible by 5 a) Write the following statements in symbolic form. i) At least one integer is even. ii) There exists a positive integer that is even. iii) If \(x\) is even, then \(x\) is not divisible by 5 . iv) No even integer is divisible by 5 . v) There exists an even integer divisible by \(5 .\) vi) If \(x\) is even and \(x\) is a perfect square, then \(x\) is divisible by 4 b) Determine whether each of the six statements in part (a) is true or falsc. For cach false statement, provide a counterexample. c) Express each of the following symbolic representations in words. i) \(\forall x[r(x) \rightarrow p(x)]\) ii) \(\forall x[s(x) \rightarrow q(x)]\) iii) \(\forall x[s(x) \rightarrow \neg t(x)]\) iv) \(\exists x[s(x) \wedge \neg r(x)]\) d) Provide a counterexample for each false statement in part (c)

17\. After baking a pie for the two nieces and two nephews who are visiting her, Aunt Nellie leaves the pie on her kitchen table to cool. Then she drives to the mall to close her boutique for the day. Upon her return she finds that someone has eaten. one-quarter of the pie. Since no one was in her house that day \(-\) except for the four visitors - Aunt Nellie questions each niece and nephew about who ate the piece of pie. The four "suspects" tell her the following: Charles: Kelly ate the piece of pie. Dawn: I did not eat the piece of pie. Kelly: Tyler ate the pic. Tyler: Kelly lied when she said I ate the pie. If only one of these four statements is true and only one of the four committed this heinous crime, who is the vile culprit that Aunt Nellie will have to punish severely?

8\. Let \(p(x), q(x)\), and \(r(x)\) denote the following open statements. $$ \begin{array}{ll} p(x): & x^{2}-8 x+15=0 \\ q(x): & x \text { is odd } \\ r(x): & x>0 \end{array} $$ For the universe of all integers, determine the truth or falsity of each of the following statements. If a statement is false, give a counterexample. a) \(\forall x[p(x) \rightarrow q(x)]\) b) \(\forall x[q(x) \rightarrow p(x)]\) c) \(\exists x[p(x) \rightarrow q(x)]\) d) \(\exists x[q(x) \rightarrow p(x)]\) e) \(\exists x[r(x) \rightarrow p(x)]\) f) \(\forall x[\neg q(x) \rightarrow \neg p(x)]\) g) \(\exists x[p(x) \rightarrow(q(x) \wedge r(x))]\) h) \(\forall x[(p(x) \vee q(x)) \rightarrow r(x)]\)

17\. Prove the following result in three ways (as in Theorem 2.4): If \(n\) is an odd integer, then \(n+11\) is even.
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