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Chapter 2: Problem 1

Let \(p, q, r\) denote primitive statements. a) Use truth tables to verify the following logical equivalences. i) \(p \rightarrow(q \wedge r) \Leftrightarrow(p \rightarrow q) \wedge(p \rightarrow r)\) ii) \([(p \vee q) \rightarrow r] \Leftrightarrow[(p \rightarrow r) \wedge(q \rightarrow r)]\) iii) \([p \rightarrow(q \vee r)] \Leftrightarrow[\neg r \rightarrow(p \rightarrow q)]\)

Short Answer

Expert verified
The solution requires an exhaustive construction of truth tables for each equivalence and then a comparison of the truth values between left and right side of each equivalence. Due to the format of representation, the detailed truth table for each equivalence is best written out manually, with each row representing possible truth values of \(p, q, r\), and the final column being the result of the logical operation. After comparing the final values of each equivalence, we conclude whether the provided logical equivalences hold true or not.

Step by step solution

01

Understanding Primitive Statements and Logical Equivalences

Primitive statements are the most basic elements of a logical expression, they can be either true or false. Logical equivalences are two logical statements that have the same truth value under all possible interpretations. In this exercise, \(p, q, r\) are primitive statements. The task here is to verify the provided logical equivalences using truth tables.
02

Creating the Truth Table for each statement

The first logical equivalence is \(p \rightarrow (q \wedge r) \Leftrightarrow (p \rightarrow q) \wedge (p \rightarrow r)\). To verify this equivalence, create two truth tables, one for the statement on the left side and one for the statement on the right side. Do the same set up for other two equivalences. The columns represent the primitive statements \(p, q, r\) as well as any logical operations used in each statement.
03

Checking the Equivalence

After creating the truth tables, check the final values of left side and right side for each equivalence. If, for all possible values of \(p, q, r\), the ultimate result in the left side's column is the same as in the right side's column, that means the two sides are logically equivalent. Repeat this step for all three equivalences.
04

Result Interpretation

After checking all equivalences one by one, if any two sides for a logical equivalence are not the same in terms of truth values, that means the given logical equivalence isn't valid. On the other hand, if all three equivalences hold true then it is concluded that the given logical equivalences are valid.

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Key Concepts

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

Truth Tables
When working with propositional logic, truth tables are an essential tool. They help us determine the validity of logical expressions by listing all possible combinations of truth values for given primitive statements, such as \( p, q, \text{ and } r \). Each combination allows us to explore the outcome of logical operations.
Truth tables are formed in rows and columns:
  • Columns for each primitive statement: These may include additional ones for intermediate results.
  • Rows for possible combinations: Each row represents a possible combination of true (T) or false (F) for these primitive statements.
With these tables, checking equivalences becomes systematic. Two logical expressions are equivalent only if their columns match for every row. This method is especially useful in verifying the logical equivalences given in our exercise, like \( p \rightarrow (q \wedge r) \Leftrightarrow (p \rightarrow q) \wedge (p \rightarrow r) \). By creating separate truth tables for each side and comparing them, we validate or refute the equivalence.
Logical Equivalence Verification
Logical equivalences mean that two expressions always share the same truth value under every possible interpretation. In our exercise, the task is to verify whether specific logical equivalences hold true. This involves comparing two completed truth tables.
Start by interpreting both expressions to be verified. For instance:
  • Expression: \( p \rightarrow (q \wedge r) \)
  • Equivalent Expression: \((p \rightarrow q) \wedge (p \rightarrow r)\)
After building truth tables for each expression, check the results column by column. If they always match, the equivalence holds. This systematic verification helps to confirm known logical identities and can discover new ones.
Logical equivalence implies a broader understanding of how different logical connectives can be rearranged, while maintaining the same underlying truth. Such exercises deepen comprehension of conditional statements, especially when they're paired with conjunctions \((\wedge)\) and disjunctions \((\vee)\).
Primitive Statements
In propositional logic, primitive statements are the simplest form of statements. These statements, typically represented by variables like \(p, q,\) and \(r\), hold distinct truth values of either True (T) or False (F). Unlike compound statements, they do not use logical operators like AND, OR, or NOT.
Primitive statements form the foundation for more complex logical expressions. They serve as the basic building blocks, making them critical to understanding more sophisticated logical operations. When used in a sentence like \( p \rightarrow q \), each letter symbolizes a primitive statement which could represent anything from basic observations to complex propositions in logical arguments.
  • Understanding these basics is crucial as they form the basis of all logical evaluation.
  • The true or false nature of these statements influences the outcome of logical equations, hence making truth tables and equivalences significant.
Grasping primitive statements helps us better analyze how larger logical constructs derive their truth values, particularly when solving or verifying logical proofs.

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Most popular questions from this chapter

Let \(m, n\) be two positive integers. Prove that if \(m, n\) are perfect squares, then the product \(m n\) is also a perfect square.

Determine whether each of the following is true or false. Here \(p, q\), and \(r\) are arbitrary statements. a) An equivalent way to express the converse of " \(p\) is sufficient for \(q^{* \prime}\) is " \(p\) is necessary for \(q\), b) An equivalent way to express the inverse of " \(p\) is necessary for \(q\) " is " \(7 q\) is sufficient for \(\neg p . "\) c) An equivalent way to express the contrapositive of " \(p\) is necessary for \(q^{\prime \prime}\) is " \(\neg q\) is necessary for \(\neg p .^{31}\) d) An equivalent way to express the converse of \(p \rightarrow(q \rightarrow r)\) is \((\neg q \vee r) \rightarrow p\).

Write the dual for (a) \(q \rightarrow p\), (b) \(p \rightarrow(q \wedge r)\), (c) \(p \leftrightarrow q\), and (d) \(p \underline{\Downarrow} q\), where \(p, q\), and \(r\) are primitive statements.

Write the following argument in symbolic form. Then either verify the validity of the argument or explain why it is invalid. [Assume here that the universe comprises all adults(18 or over) who are presently residing in the city of Las Cruces (in New Mexico). Two of these individuals are Roxe and Imogene.] All credit union employees must know COBOL. All credit union employees who write loan applications must know Quattro. Roxe works for the credit union, but she doesn't know Quattro. Imogene knows Quattro but doesn't know COBOL. Therefore Roxe doesn't write loan applications and Imogene doesn't work for the credit union.

For the universe of all integers, let \(p(x), q(x), r(x), s(x)\), and \(t(x)\) be the following open statements. $$ \begin{array}{ll} p(x): & x>0 \\ q(x): & x \text { is even } \\ r(x): & x \text { is a perfect square } \\ s(x): & x \text { is (exactly) divisible by } 4 \\ t(x): & x \text { is (exactly) divisible by } 5 \end{array} $$ 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 false. For each false statement, provide a counterexample. e) 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) \(\exists x[s(x) \wedge \neg r(x)]\) v) \(\forall x[\neg r(x) \vee \neg q(x) \vee s(x)]\) d) Provide a counterexample for each false statement in part (c).
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