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Chapter 2: Problem 1

1\. Let \(p(x), q(x)\) denote the following open statements. \(p(x): \quad x \leq 3 \quad q(x): \quad x+1\) is odd f the universe consists of all integers, what are the truth values of the following statements? a) \(q(1)\) b) \(\neg p(3)\) c) \(p(7) \vee q(7)\) d) \(p(3) \wedge q(4)\) e) \(\neg(p(-4) \vee q(-3))\) f) \(\neg p(-4) \wedge \neg q(-3)\)

Short Answer

Expert verified
a) False, b) False, c) True, d) True, e) False, f) False.

Step by step solution

01

Define predicates

Define \(p(x)\) as \(x \leq 3\) and \(q(x)\) as '\(x+1\)' is odd, where the universe is all integers.
02

Evaluate statement a)

To evaluate \(q(1)\), substitute \(1\) in \(q(x)\), which results in \(2\). Since 2 is not an odd number, \(q(1)\) is false.
03

Evaluate statement b)

To evaluate \(\neg p(3)\), remember \(\neg\) stands for negation, meaning the statement \(p(3)\) has to be true for \(\neg p(3)\) to be false and vice versa. Substituting 3 into \(p(x)\) yields true since 3 is less than or equal to 3. Therefore, \(\neg p(3)\) is false.
04

Evaluate statement c)

To evaluate \(p(7) \vee q(7)\), remember \(\vee\) stands for 'or'. This means the statement is true if either or both of \(p(7)\) or \(q(7)\) are true. Substituting 7 into \(p(x)\), we obtain false. But substituting 7 into \(q(x)\) yields true since 8 is an even number. Hence, \(p(7) \vee q(7)\) is true.
05

Evaluate statement d)

To evaluate \(p(3) \wedge q(4)\), remember \(\wedge\) stands for 'and'. Thus the statement is true only if both \(p(3)\) and \(q(4)\) are true. Substituting 3 into \(p(x)\) and 4 into \(q(x)\), both yield true. Therefore, \(p(3) \wedge q(4)\) is true.
06

Evaluate statement e)

\(\neg(p(-4) \vee q(-3))\) is a negation of the statement \(p(-4) \vee q(-3)\). Substituting \(-4\) into \(p(x)\) and \(-3\) into \(q(x)\) results in true and false respectively. Hence, \(p(-4) \vee q(-3)\) is true and its negation \(\neg(p(-4) \vee q(-3))\) is false.
07

Evaluate statement f)

To evaluate \(\neg p(-4) \wedge \neg q(-3)\), remember \(\wedge\) signifies 'and' while \(\neg\) signifies negation. The statement is true only if both \(\neg p(-4)\) and \(\neg q(-3)\) are true. However, substituting \(-4\) into \(p(x)\) results in true, hence \(\neg p(-4)\) is false. Consequently, it doesn't matter what value \(\neg q(-3)\) yields, the entire statement \(\neg p(-4) \wedge \neg q(-3)\) is false.

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Key Concepts

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

Logical Connectives
In the realm of logic, logical connectives are the glue that holds complex statements together. They're operators that connect statements, or 'predicates', to form more elaborate propositions which may be subject to truth-value evaluation. Some of the most fundamental logical connectives include AND (symbolized as \(\wedge\)), OR (symbolized as \(\vee\)), and NOT (symbolized as \(eg\)).

Each connective has its own rules for assigning truth values to the combined statements. For instance, an 'AND' (\(\wedge\)) statement is true only if both combined predicates are true, while an 'OR' (\(\vee\)) statement is true if at least one of the predicates is true. The 'NOT' (\(eg\)) operator simply inverts the truth value of a single predicate.

Understanding these connectives is crucial for analyzing logical arguments and assessing the validity of propositions within predicate logic. As you've seen in the exercise, each logical connective modulates the truth value of the combined predicates, thus altering the overall truth of the statement.
Truth Value Evaluation
The process of truth value evaluation involves determining whether a statement or a proposition is true or false. In predicate logic, this entails looking at the components of the statements (predicates and individual constants) and understanding how they interact based on the logical connectives between them.

In practice, you often start by assessing the truth of the individual predicates, like \(p(x)\) and \(q(x)\) from our exercise. Once you establish their truth values for specific values of 'x', you then apply the rules of the logical connectives to come to a final conclusion about the truth of the overall statement.

The exercise expertly demonstrates the step-by-step evaluation of truth values. Remember, to properly evaluate each statement, keep track of the universe of discourse (in this case, the integers) and apply the definitions of the predicates combined with the logical connectives.
Predicate Calculus
At the heart of more complex logical reasoning lies predicate calculus, also known as predicate logic. It extends propositional logic by dealing with predicates, which can express properties about objects, and quantifiers, which can indicate the scope of these properties over a domain of discourse.

In predicate calculus, every predicate has an associated truth value that can depend on the variables within it. Different variables can lead to different truth values, as seen in the exercise where the truth of \(p(x)\) and \(q(x)\) changes with different integer substitutions for 'x'. Predicate calculus is a powerful tool for formalizing mathematical proofs and reasoning about statements in a rigorous way.

Learning to work with predicate calculus involves recognizing the structure of predicates, the role of logical connectives, and understanding the domains over which the predicates range. By mastering the use of predicates and logical operators, you can tackle complex logical problems and develop a strong foundation for further study in logic and mathematics.

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Most popular questions from this chapter

21\. For the following statements the universe comprises all nonzero integers. Determine the truth value of each statement. a) \(\exists x \exists y[x y=1]\) b) \(\exists x \forall y[x y=1]\) c) \(\forall x \exists y[x y=1]\) d) \(\exists x \exists y[(2 x+y=5) \wedge(x-3 y=-8)]\) e) \(\exists x \exists y[(3 x-y=7) \wedge(2 x+4 y=3)]\)

3\. Let \(p, q\), and \(r\) denote primitive statements. Prove or disorove (provide a counterexample for) each of the following. a) \([p \leftrightarrow(q \leftrightarrow r)] \Longleftrightarrow[(p \leftrightarrow q) \leftrightarrow r]\) b) \([p \rightarrow(q \rightarrow r)] \Longleftrightarrow[(p \rightarrow q) \rightarrow r]\)

11\. a) How many rows are needed for the truth table of the compound statement \((p \vee \neg q) \leftrightarrow[(\neg r \wedge s) \rightarrow t]\), where \(p, q, r, s\), and \(t\) are primitive statements? b) Let \(p_{1}, p_{2}, \ldots, p_{n}\) denote \(n\) primitive statements. Let \(p\) be a compound statement that contains at least one occurrence each of \(p_{1}\), for \(1 \leq i \leq n-\) and \(p\) contains no other primitive statement. How many rows are needed to construct the truth table for \(p ?\)

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?

19\. For each of the following statements state the converse, inverse, and contrapositive. Also determine the truth value for each given statement, as well as the truth values for its converse, inverse, and contrapositive. (Here "divides" means "exactly divides.") a) [The universe comprises all positive integers.] If \(m>n\), then \(m^{2}>n^{2}\) b) [The universe comprises all integers.] If \(a>b\), then \(a^{2}>b^{2}\). c) [The universe comprises all integers.] If \(m\) divides \(n\) and \(n\) divides \(p\), then \(m\) divides \(p\). d) [The universe consists of all real numbers.] \(\forall x\left[(x>3) \rightarrow\left(x^{2}>9\right)\right]\) e) [The universe consists of all real numbers.] For all real numbers \(x\), if \(x^{2}+4 x-21>0\), then \(x>3\) or \(x<-7\)
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