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Chapter 2: Problem 6

Let \(p, q, r\) denote primitive statements. Write the converse, inverse, and contrapositive of each of the following implications. a) \(p \rightarrow(q \wedge r)\) b) \((p \vee q) \rightarrow r\)

Short Answer

Expert verified
For the implication \(p \rightarrow (q \wedge r)\): \n- The converse is \((q \wedge r) \rightarrow p\); \n- The inverse is \(\sim p \rightarrow (\sim q \vee \sim r)\); \n- The contrapositive is \((\sim q \vee \sim r) \rightarrow \sim p\). \n\n For the implication \((p \vee q) \rightarrow r\): \n- The converse is \(r \rightarrow (p \vee q)\); \n- The inverse is \((\sim p \wedge \sim q) \rightarrow \sim r\); \n- The contrapositive is \(\sim r \rightarrow (\sim p \wedge \sim q)\).

Step by step solution

01

Find converse, inverse, and contrapositive for the first implication

Let's consider the first implication \(p \rightarrow (q \wedge r)\). The converse of this implication will be \((q \wedge r) \rightarrow p\). The inverse of this implication will be \(\sim p \rightarrow \sim (q \wedge r)\), which simplifies to \(\sim p \rightarrow (\sim q \vee \sim r)\) using De Morgan's law. The contrapositive of this implication will be \(\sim (q \wedge r) \rightarrow \sim p\) which simplifies to \( (\sim q \vee \sim r) \rightarrow \sim p\) using De Morgan's law.
02

Find converse, inverse, and contrapositive for the second implication

Now let's consider the second implication \((p \vee q) \rightarrow r\). The converse of this implication will be \(r \rightarrow (p \vee q)\). The inverse of this implication will be \(\sim (p \vee q) \rightarrow \sim r\), which simplifies to \((\sim p \wedge \sim q) \rightarrow \sim r\) using De Morgan's law. The contrapositive of this implication will be \(\sim r \rightarrow \sim (p \vee q)\), which simplifies to \(\sim r \rightarrow (\sim p \wedge \sim q)\) using De Morgan's law.

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Key Concepts

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

Understanding the Converse
In the world of logic, the converse is an important concept to grasp. When you have a logical implication of the form \( p \rightarrow q \), its converse would be \( q \rightarrow p \). Essentially, we're flipping the direction of the implication. This does not necessarily mean the converse is true if the original statement is true.

To illustrate with an example: If we have the implication \( p \rightarrow (q \wedge r) \), the converse is \( (q \wedge r) \rightarrow p \). Here, we are saying that if both \( q \) and \( r \) are true, then \( p \) must be true. Similarly, for \( (p \vee q) \rightarrow r \), the converse is \( r \rightarrow (p \vee q) \). This means that if \( r \) is true, then either \( p \) or \( q \) must be true.

Key points about converse:
  • The converse reverses the conditional statement.
  • Truth of the converse does not guarantee the truth of the original statement.
  • Useful in exploring logical relationships and conclusions.
Exploring the Inverse
The inverse of a logical statement is another fundamental concept. An inverse negates both the hypothesis and the conclusion of the original implication. If the original implication is \( p \rightarrow q \), the inverse would be \( \sim p \rightarrow \sim q \), where \( \sim \) denotes negation.

For practical understanding, consider the statement \( p \rightarrow (q \wedge r) \). The inverse becomes \( \sim p \rightarrow \sim (q \wedge r) \), which simplifies to \( \sim p \rightarrow (\sim q \vee \sim r) \) using De Morgan's law. This means that if \( p \) is not true, then at least one of \( q \) or \( r \) is also not true.

Similarly, for the implication \( (p \vee q) \rightarrow r \), the inverse is \( \sim (p \vee q) \rightarrow \sim r \), simplifying to \( (\sim p \wedge \sim q) \rightarrow \sim r \). This indicates that if neither \( p \) nor \( q \) is true, \( r \) is also not true.

Important aspects of inverse:
  • Negates both the hypothesis and the conclusion.
  • Like the converse, the truth of the inverse does not suggest the truth of the original.
  • Useful in contradiction and testing logical boundaries.
Delving into Contrapositive
The contrapositive is closely related to the original implication and is logical equivalent to it, meaning both statements share the same truth value. When you have \( p \rightarrow q \), the contrapositive is \( \sim q \rightarrow \sim p \).

For example, if the original implication is \( p \rightarrow (q \wedge r) \), the contrapositive is \( \sim (q \wedge r) \rightarrow \sim p \), which simplifies to \( (\sim q \vee \sim r) \rightarrow \sim p \). This suggests that if not both \( q \) and \( r \) are true, then \( p \) cannot be true.

Similarly, for \( (p \vee q) \rightarrow r \), the contrapositive is \( \sim r \rightarrow \sim (p \vee q) \), simplifying to \( \sim r \rightarrow (\sim p \wedge \sim q) \). It reflects that if \( r \) is not true, neither \( p \) nor \( q \) can be true.

Key takeaways on contrapositive:
  • Inverse of the converse of the original statement.
  • Always shares the same truth value as the original implication.
  • A powerful tool for proving logical statements.

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Most popular questions from this chapter

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 any statements \(p, q\), prove that a) \(\neg(p \downarrow q) \Leftrightarrow(\neg p \uparrow \neg q)\) b) \(\neg(p \uparrow q) \Leftrightarrow(\neg p \downarrow \neg q)\)

a) Let \(p(x, y)\) denote the open statement " \(x\) divides \(y\)," where the universe for each of the variables \(x, y\) comprises all integers. (In this context "divides" means "exactly divides" or "divides evenly.") Determine the truth value of each of the following statements; if a quantified statement is false, provide an explanation or a counterexample. i) \(p(3,7)\) ii) \(p(7,3)\) iii) \(p(3,27)\) iv) \(\forall y p(1, y)\) v) \(\forall x p(x, 0)\) vi) \(\forall x p(x, x)\) vii) \(\forall y \exists x p(x, y)\) viii) \existsy \(\forall x p(x, y)\) ix) \(\forall x \forall y[(p(x, y) \wedge p(y, x)) \rightarrow(x=y)]\) x) \(\forall x \forall y \forall z[(p(x, y) \wedge p(y, z)) \rightarrow p(x, z)]\) b) Determine which of the 10 statements in part (a) will change in truth value if the universe for each of the variables \(x, y\) were restricted to just the positive integers. c) Determine the truth value of each of the following statements. If the statement is false, provide an explanation or a counterexample. [The universe for each of \(x, y\) is as in part (b).] i) \(\forall x \exists y p(x, y)\) ii) \(\forall y \exists x p(x, y)\) iii) \(\exists x \forall y p(x, y)\) iv) \(\exists y \forall x p(x, y)\)

Determine whether each of the following pairs of statements is logically equivalent. [Here \(p, q, r\) are primitive statements and \((p \downarrow q) \Leftrightarrow \neg(p \vee q)\) while \((p \uparrow q) \Leftrightarrow \neg(p \wedge q) .]\) a) \(p \downarrow(q \downarrow r)\), b) \(p \uparrow(q \downarrow r), \quad(p \uparrow q) \downarrow(p \uparrow r)\) c) \(p \downarrow(q \uparrow r), \quad(p \downarrow q) \uparrow(p \downarrow r)\)

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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