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Chapter 2: Problem 16

For primitive statements \(p, q\), a) verify that \(p \rightarrow[q \rightarrow(p \wedge q)]\) is a tautology. b) verify that \((p \vee q) \rightarrow[q \rightarrow q]\) is a tautology by using the result from part (a) along with the substitution rules and the laws of logic. c) is \((p \vee q) \rightarrow[q \rightarrow(p \wedge q)]\) a tautology?

Short Answer

Expert verified
a) Yes, \(p \rightarrow[q \rightarrow(p \land q)]\) is a tautology. b) Yes, by using the result of a, along with substitution and logical laws, we can verify that \((p \vee q) \rightarrow[q \rightarrow q]\) is a tautology. c) Needs to be verified by creating a truth table.

Step by step solution

01

Understand the Problem

The problem consists of verifying if the given three primitive statements are tautologies. This involves constructing truth tables for each statement.
02

Verify the first expression

Start by creating a truth table for \(p \rightarrow[q \rightarrow(p \wedge q)]\). The outcome should result in all true final column, proving it's a tautology.
03

Verify the second expression

Use the result from step 2, along with substitution rules and laws of logic, to prove that \((p \vee q) \rightarrow[q \rightarrow q]\) is a tautology. The truth table for this will also result in a 'true' final column.
04

Verify the third expression

Finally, create the truth table for the expression \((p \vee q) \rightarrow[q \rightarrow(p \wedge q)]\). By doing so, we will ultimately find out whether this is a tautology or not.

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Key Concepts

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

Truth Tables
Truth tables are a foundational tool in the study of logic, used to determine the truth value of logical expressions based on the truth values of their components. These tables list all possible combinations of true (T) and false (F) values for the variables involved and show how these values combine under various logical operations. To understand tautologies, students will often construct truth tables and look for a column of resulting values that consists entirely of 'T', which indicates that the expression will always be true, no matter the truth values of its individual components. This characteristic is essential for recognizing tautologies, as they are statements that are true by virtue of their logical form alone.

For example, when evaluating the expression \(p \rightarrow[q \rightarrow(p \wedge q)]\) using a truth table, we would systematically work through all combinations of truth values for \(p\) and \(q\). If each possible combination results in the expression being true, we have confirmed that the expression is a tautology.
Laws of Logic
The laws of logic are rules that govern the combination of truth values in logical statements. These include principles such as the Law of Identity, which states that an expression is identical to itself; the Law of Noncontradiction, asserting that a statement and its negation cannot both be true; and the Law of Excluded Middle, which indicates that a statement must either be true or false, with no middle ground. Building upon these foundational laws are the rules for logical operations like 'and' (\(\wedge\)), 'or' (\(\vee\)), and 'implies' (\(\rightarrow\)).

When solving logical expressions, such as proving a tautology, students may use substitution rules and these logical laws to simplify expressions. For instance, the expression from part b \(\) of the exercise, \(\), can be broken down and simplified using these laws, making it easier to analyze. Knowing these principles is essential for understanding how to manipulate and interpret logical statements.
Logical Expressions
Logical expressions, or propositional expressions, are sentences that express a statement that can be evaluated as true or false. These expressions are usually composed of propositions represented by variables (like \(p\) and \(q\) in our exercise), logical connectives (like \(\wedge\), \(\vee\), and \(\rightarrow\)), and possibly parentheses to indicate the order of operations. When analyzing logical expressions, it鈥檚 important to understand how the values of the propositions influence the overall value of the expression.

For example, in an implication \(p \rightarrow q\), the truth of \(q\) is guaranteed if \(p\) is true. However, if \(p\) is false, the entire expression is true regardless of \(q\)'s truth value. This counterintuitive aspect of implication is often a stumbling point for students first learning about logical expressions. Strengthening one's grasp of logical expressions and how they combine is critical for anyone delving into the realm of logical reasoning and mathematics.

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Most popular questions from this chapter

Write each of the following arguments in symbolic form. Then either establish the validity of the argument or provide a counterexample to show that it is invalid. a) If it is cool this Friday, then Craig will wear his suede jacket if the pockets are mended. The forecast for Friday calls for cool weather, but the pockets have not been mended. Therefore Craig won't be wearing his suede jacket this Friday. b) The contract will be fulfilled if and only if the new windows are installed in the house in June. If the new windows are installed in June, then Evelyn can move into her new house on the first of July. If she can't move in on July 1 , then Evelyn must pay the July rent on her apartment. The windows have been installed in June or Evelyn must pay the July rent on her apartment. Therefore Evelyn won't have to pay rent on her apartment for July.

Use truth tables to verify that each of the following is a logical implication. a) \([(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)\) b) \([(p \rightarrow q) \wedge \neg q] \rightarrow \neg p\) c) \([(p \vee q) \wedge \neg p] \rightarrow q\) d) \([(p \rightarrow r) \wedge(q \rightarrow r)] \rightarrow[(p \vee q) \rightarrow r]\)

For each of the following statements provide an indirect proof [as in part (2) of Theorem 2.4] by stating and proving the contrapositive of the given statement. a) For all integers \(k\) and \(l\), if \(k l\) is odd, then \(k, l\) are both odd. b) For all integers \(k\) and \(l\), if \(k+l\) is even, then \(k\) and \(l\) are both even or both odd.

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).

Let the universe for the variables in the following statements consist of all real numbers. In each case negate and simplify the given statement. a) \(\forall x \forall y[(x>y) \rightarrow(x-y>0)]\) b) \(\forall x \forall y\left[\left[(x>0) \wedge\left(y=\log _{10} x\right)\right] \rightarrow\left(x=10^{y}\right)\right]\) c) \(\forall x \forall y[(x0) \wedge(y>0))] \rightarrow[\exists z(x z>y)]\)
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