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Chapter 2: Problem 15

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 .

Short Answer

Expert verified
a. True b. True c. False, counterexample: \(16 > 0\) d. True e. False, counterexample: \(36\) has a tens digit of \(3\)

Step by step solution

01

Statement a: 鈭�x 鈭� D, if x is odd then x>0.

For each element in the set D, check if the element is odd and greater than 0. Elements in D that are odd: {-1, 3, 23}. All these odd elements are greater than 0, so statement a is true.
02

Statement b: 鈭�x 鈭� D, if x is less than 0 then x is even.

For each element in the set D, check if the element is less than 0 and if it is even. Elements in D that are less than 0: {-48, -14, -8}. All these elements are even, so statement b is true.
03

Statement c: 鈭�x 鈭� D, if x is even then x 鈮� 0.

For each element in the set D, check if the element is even and less than or equal to 0. Elements in D that are even: {-48, -14, -8, 0, 16, 26, 32, 36}. In this case, we can see that some of the even elements (16, 26, 32, 36) are greater than 0, making statement c false. A counterexample is 16, which is an even element and greater than 0.
04

Statement d: 鈭�x 鈭� D, if the ones digit of x is 2, then the tens digit is 3 or 4.

For each element in the set D, check if the ones digit is 2 and if the tens digit is either 3 or 4. Elements in D with a ones digit of 2: {32}. 32 has a tens digit of 3, so statement d is true.
05

Statement e: 鈭�x 鈭� D, if the ones digit of x is 6, then the tens digit is 1 or 2.

For each element in the set D, check if the ones digit is 6 and if the tens digit is either 1 or 2. Elements in D with a ones digit of 6: {26, 36}. 26 has a tens digit of 2 and 36 has a tens digit of 3. In this case, the statement is false since 36 has a tens digit of 3, which is not 1 or 2 as claimed by the statement. A counterexample is 36, which has a ones digit of 6 and a tens digit of 3.

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Key Concepts

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

Quantifiers
In discrete mathematics, quantifiers play a crucial role in the formal expression of statements involving logic. They allow us to make generalizations about elements within a set. There are two main types of quantifiers:
  • Universal Quantifier (\( \forall \)): This quantifier is used to state that a proposition is true for all elements in a particular set. For example, "\( \forall x \in D\), if \( x \) is odd then \( x>0 \)" means this statement must hold true for every element \( x \) in the set \( D \).
  • Existential Quantifier (\( \exists \)): It indicates that there exists at least one element in the set for which the proposition is true. This is not as common in the given exercise, but knowing it helps in understanding logic statements.
By using quantifiers, mathematicians can express complex logical conditions concisely.
Counterexample
A counterexample is a critical concept in proving mathematical statements, especially those involving universal quantifiers. It serves as an example that disproves a general statement. When we say a statement is false, providing a counterexample is an essential step. Let's look at the process:
  • Identify the Assertion: We start by understanding the statement and identifying the condition it is asserting, such as "for all elements, a specific property holds true."
  • Search for a Violation: We then check each element (or a sample, if feasible) in the specified set to find an instance that does not satisfy the statement's condition.
  • Present the Counterexample: Once an element violating the condition is identified, it is presented as the counterexample that disproves the universal statement. For example, in statement 'e' of the provided exercise, 36 is a counterexample since it doesn't fit the rule that the tens digit should be 1 or 2 if the ones digit is 6.
Counterexamples are powerful tools in the realm of logic, as they can immediately show the falsehood of a statement asserted to be universally true.
Truth Values
Truth values are fundamental when dealing with logical statements in mathematics. They help us determine whether a statement is indeed true or false. For each logical statement in the exercise, you assess the truth value based on the condition and context provided by the quantifier. The process of determining truth values involves:
  • Evaluation: Examine each proposition carefully (e.g., "\( x \) is odd then \( x>0 \)"). Ensure that the condition holds for each element in this context.
  • Consistency Check: For statements involving the universal quantifier (\( \forall \)), verify that no single element contradicts the statement, as even a single counterexample makes it false. For statements like "\( \forall x \in D\), if \( x \) is even then \( x \leq 0 \)", finding elements 16, 26, 32, 36 which do not satisfy the condition immediately assigns it false.
  • Recording Results: Once evaluated, allocate the final truth value (True or False) to the statement. This helps in understanding the logical conclusion drawn from the data set.
Truth values are essential in differentiating valid logical constructs from false ones, enhancing our logical reasoning skills.

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Most popular questions from this chapter

Consider the following string of numbers: 0204. A person claims that all the 1's in the string are to the left of all the 0 's in the string. Is this true? Justify your answer. (Hint: Write the claim formally and write a formal negation for it. Is the negation true or false?)

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.

Consider the following statement: \(\forall\) basketball players \(x, x\) is tall.

Use universal modus tollens to fill in valid conclusions for the arguments in 5 and 6 . All healthy people eat an apple a day. Adster does not eat an apple a day.

In exercises \(28-33\), reorder the premises in each of the arguments to show that the conclusion follows as a valid consequence from the premises. It may be helpful to rewrite the statements in ifthen form and replace some statements by their contrapositives. Exercises 28-30 refer to the kinds of Tarski worlds discussed in Example 2.1.12 and 2.3.1. Exercises 31 and 32 are adapted from Symbolic Logic by Lewis Carroll.* 1\. I trust every animal that belongs to me. 2\. Dogs gnaw bones. 3\. I admit no animals into my study unless they will beg when told to do so. 4\. All the animals in the yard are mine. 5\. I admit every animal that I trust into my study. 6\. The only animals that are really willing to beg when told to do so are dogs. \(\therefore\) All the animals in the yard gnaw bones.
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