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Chapter 2: Problem 37

Give an example to show that a universal conditional statement is not logically equivalent to its inverse.

Short Answer

Expert verified
Consider the propositions P(x): "x is even" and Q(x): "x is divisible by 4". The universal conditional statement is \(S_1:\) "For all x, if x is even then x is divisible by 4". The inverse statement is \(S_2:\) "For all x, if x is not even then x is not divisible by 4". Let x = 6; \(S_1\) is false and \(S_2\) is true, showing that \(S_1\) and \(S_2\) are not logically equivalent.

Step by step solution

01

Define the propositions P(x) and Q(x)

Let's take P(x) to be "x is even", and Q(x) to be "x is divisible by 4". We will use these propositions to form both a universal conditional statement and its inverse.
02

Form the universal conditional statement

Using the propositions P(x) and Q(x), we can form a universal conditional statement "For all x, if x is even (P(x)) then x is divisible by 4 (Q(x))". Let's represent this statement as \(S_1\): \(S_1:\) "For all x, if x is even then x is divisible by 4".
03

Form the inverse statement

To create the inverse of the universal conditional statement, we need to negate both P(x) and Q(x), resulting in the statement "For all x, if x is not even then x is not divisible by 4". Let's represent this statement as \(S_2\): \(S_2:\) "For all x, if x is not even then x is not divisible by 4".
04

Test the statements for logical equivalence

To check if the statements are logically equivalent, we will see if there exists a counterexample where both statements have different truth values. Let x = 6. It is an even number but it is not divisible by 4. - In the case of \(S_1\), since x is even (P(x) is true) but not divisible by 4 (Q(x) is false), this makes the statement "If x is even then x is divisible by 4" false. - In the case of \(S_2\), since x is even (not P(x) is false) and not divisible by 4 (not Q(x) is true), the statement "If x is not even then x is not divisible by 4" is true (due to the behavior of the conditional, "false implies true" is true). Thus, we have shown a counterexample where the universal conditional statement \(S_1\) is not logically equivalent to its inverse \(S_2\), as they have different truth values for the same input (x = 6).

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Key Concepts

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

Logical Equivalence
Understanding logical equivalence is crucial in the study of formal logic and reasoning. Two statements are said to be logically equivalent if, and only if, they have the same truth value in every possible scenario. This means that no matter what the circumstances or the values being substituted into the statements, both statements will either be true or false together.

For a straightforward example, consider the two statements: 'If it rains, the ground is wet,' and 'If the ground is not wet, then it did not rain.' Assuming that there are no other factors making the ground wet, these statements are logically equivalent because they bear the same implication about the relationship between rain and the wetness of the ground. This is generally tested for using truth tables or by demonstrating that one statement can be derived from another using valid logical steps.
Inverse Conditional Statement
In contrast to logical equivalence, an inverse conditional statement results from negating both the hypothesis and conclusion of an original conditional statement. For example, if the original statement is 'If the weather is cold (P), then I will wear a coat (Q),' its inverse is 'If the weather is not cold (not P), then I will not wear a coat (not Q).'

It's essential to note that an original conditional statement is not always logically equivalent to its inverse. The truth of one does not guarantee the truth of the other, as the relationship between the hypothesis and conclusion may not be bidirectional. This is why the exercise to provide a counterexample can be so useful - it clearly demonstrates the lack of logical equivalence between a statement and its inverse by identifying a scenario in which the two differ in truth value.
Counterexample
The use of a counterexample is a powerful technique in mathematical logic to disprove a claim, such as the supposed logical equivalence between two statements. By finding just one instance in which the claim does not hold, the argument can be invalidated.

In our exercise, the number 6 serves as a counterexample to disprove the logical equivalence between the universal conditional statement 'If x is even then x is divisible by 4' and its inverse 'If x is not even then x is not divisible by 4.' The number 6 is even, which satisfies the hypothesis of the first statement but does not meet the criterion of being divisible by 4, rendering the first statement false. However, the inverse statement does not even consider the number 6, as it only deals with numbers that are not even, demonstrating that the truth conditions for both statements are not the same.

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Most popular questions from this chapter

Consider the following statement: \(\forall\) integers \(n\), if \(n^{2}\) is even then \(n\) is even. Which of the following are equivalent ways of expressing this statement? a. All integers have even squares and are even. b. Given any integer whose square is even, that integer is itself even. c. For all integers, there are some whose square is even. d. Any integer with an even square is even. e. If the square of an integer is even, then that integer is even. f. All even integers have even squares.

Let \(D=\\{-48,-14,-8,0,1,3,16,23,26,32,36\\}\). Determine which of the following statements are true and which are false. Provide counterexamples for those state ments that are false. a. \(\forall x \in D\), if \(x\) is odd then \(x>0\). b. \(\forall x \in D\), if \(x\) is less than 0 then \(x\) is even. c. \(\forall x \in D\), if \(x\) is even then \(x \leq 0\). d. \(\forall x \in D\), if the ones digit of \(x\) is 2 , then the tens digit is 3 or 4 . e. \(\forall x \in D\), if the ones digit of \(x\) is 6 , then the tens digit is 1 or 2 .

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Which of the following is a negation for "All dogs are loyal"? More than one answer may be correct. a. All dogs are disloyal. b. No dogs are loyal. c. Some dogs are disloyal. d. Some dogs are loyal. e. There is a disloyal animal that is not a dog. f. There is a dog that is disloyal. g. No animals that are not dogs are loyal. \(\mathrm{h}\). Some animals that are not dogs are loyal.

Rewrite each of the following statements in the form \(" v\) \(x\), if then \(x\) and \(y\), if then \("\) or " a. All Java programs have at least 5 lines. b. Any valid argument with true premises has a true conclusion. c. The sum of any two even integers is even. d. The product of any two odd integers is odd.
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