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Chapter 1: Problem 9

Indicate whether each statement is True or False. t? (a) \(3 \leq 5\) and 11 is odd. (b) \(3^{2}=8\) or \(2+3=5\). (c) \(5>8\) or 3 is even. (d) If 6 is even, then 9 is odd. (e) If \(8<3\), then \(2^{2}=5\). (f) If 7 is odd, then 10 is prime. (g) If 8 is even and 5 is not prime, then \(4<7\). (h) If 3 is odd or \(4>6\), then \(9 \leq 5\). (i) If both \(5-3=2\) and \(5+3=2\), then \(9=4\). (j) It is not the case that 5 is even or 7 is prime.

Short Answer

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True, True, False, True, True, False, False, True, False, False

Step by step solution

01

Statement (a)

(a) \(3 \leq 5\) and 11 is odd. Both parts here are true. \(3 \leq 5\) is true because 3 is less than or equal to 5. Additionally, 11 is odd as there is no integer that can be multiplied by 2 to equal 11. Since both parts are true, the overall statement is True.
02

Statement (b)

(b) \(3^{2}=8\) or \(2+3=5\). In this statement, the first part (\(3^2 = 8\)) is false because \(3^2 = 9\). However, the second part (\(2 + 3 = 5\)) is true. Since this statement is an "or" statement, the entire statement is true if at least one part is true. Therefore, the overall statement is True.
03

Statement (c)

(c) \(5>8\) or 3 is even. Both parts here are false. \(5 > 8\) is false, and 3 is odd, not even. Since both parts are false, the overall statement is False.
04

Statement (d)

(d) If 6 is even, then 9 is odd. This is a conditional statement. 6 is indeed even, and 9 is also odd. Since the conditions are satisfied, the overall statement is True.
05

Statement (e)

(e) If \(8<3\), then \(2^{2}=5\). This is another conditional statement. However, the first part (\(8 < 3\)) is false. In a conditional statement, if the first part is false, the whole statement is automatically considered true, regardless of the second part. Therefore, the overall statement is True.
06

Statement (f)

(f) If 7 is odd, then 10 is prime. 7 is indeed odd. However, 10 is not prime because it has more than two factors (1, 2, 5, and 10). Since the conditions in the statement are not satisfied, the overall statement is False.
07

Statement (g)

(g) If 8 is even and 5 is not prime, then \(4<7\). This is another conditional statement. The first condition (if 8 is even) is true, but the second condition (that 5 is not prime) is false. Since both conditions are not satisfied, we don't need to examine the second part of the statement, and the overall statement is False.
08

Statement (h)

(h) If 3 is odd or \(4>6\), then \(9 \leq 5\). The first part (3 is odd) is true, making the entire conditional statement True regardless of the second part. However, even if we consider the second part (\(9 \leq 5\)), which is false, the overall statement is still True because the first condition was met.
09

Statement (i)

(i) If both \(5-3=2\) and \(5+3=2\), then \(9=4\). The first part (\(5 - 3 = 2\)) is true, but the second part (\(5 + 3 = 2\)) is false. Since both conditions are not satisfied, we don't need to examine the second part of the statement, and the overall statement is False.
10

Statement (j)

(j) It is not the case that 5 is even or 7 is prime. The first part of this statement (5 is even) is false, but the second part (7 is prime) is true. Since there is a true statement within the "or" statement, the "not" in front of the statement makes the overall statement False.

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Key Concepts

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

Conditional Statements
Conditional statements form a significant part of logic in mathematics. These statements often follow a simple "if-then" structure, like the statements "If 6 is even, then 9 is odd," or "If 8 is even and 5 is not prime, then 4 < 7." In these statements:
  • The "if" part is known as the hypothesis or antecedent.
  • The "then" part is called the conclusion or consequent.
The entire statement is considered true unless a true hypothesis leads to a false conclusion.
For instance, the statement "If 7 is odd, then 10 is prime" is false because even though 7 is odd, 10 is not prime, thus not satisfying the conclusion.
Truth Values
Truth values in logic determine whether each component of a statement is true or false. This applies to simple assertions like "11 is odd" and complex conditional statements like "If both 5-3=2 and 5+3=2, then 9=4." In a mathematical logic context:
  • An assertion is true if the information it presents is factual.
  • An assertion is false if it conflicts with known truths.
When handling "or" statements, as in "3虏=8 or 2+3=5," remember that the statement is true if at least one part is true.
On the other hand, "and" statements require all parts to be true, exemplified in "3鈮�5 and 11 is odd," where both must be verified as true.
Mathematical Reasoning
Mathematical reasoning is the process used to determine the truth value of various statements. It requires analyzing each component systematically, often starting from known truths or axioms. Here's how it works:
  • Identify the statements or propositions to evaluate.
  • Break complex statements into simpler parts, such as hypotheses and conclusions in conditionals.
  • Apply logical rules, such as the truth tables for "and," "or," and "if-then" (conditional) statements.
For example, to verify "If 6 is even, then 9 is odd," note that 6 being even and 9 being odd are both true, confirming the statement through direct evaluation. Mathematical reasoning equips students to logically deduce truth values and enhances their overall understanding of mathematical assertions.

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Most popular questions from this chapter

Identify the antecedent and the consequent in each statement. (a) A sequence is convergent if it is Cauchy. (b) Convergence is a necessary condition for boundedness. (c) Orthogonality implies invertability. (d) \(K\) is closed and bounded only if \(K\) is compact.

Construct a truth table for each statement. (a) \(p \vee \sim q\) (b) \(p \wedge \sim p\) (c) \([(\sim q) \wedge(p \Rightarrow q)] \Rightarrow \sim p\)

Mark each statement True or False. Justify each answer. (a) In order to be classified as a statement, a sentence must be true. (b) Some statements are both true and false. (c) When statement \(p\) is true, its negation \(\sim p\) is false. (d) A statement and its negation may both be false. (e) In mathematical logic, the word "or" has an inclusive meaning.

Write the negation of each statement. it (a) The \(3 \times 3\) identity matrix is singular. (b) The function \(f(x)=\sin x\) is bounded on \(\mathbb{R}\). (c) The functions \(f\) and \(g\) are linear. (d) Six is prime or seven is odd. (e) If \(x\) is in \(D\), then \(f(x)<5\). (f) If \(\left(a_{n}\right)\) is monotone and bounded, then \(\left(a_{n}\right)\) is convergent. (g) If \(f\) is injective, then \(S\) is finite or denumerable.

Identify the antecedent and the consequent in each statement. (a) A sequence is convergent if it is Cauchy. (b) Convergence is a necessary condition for boundedness. (c) Orthogonality implies invertability. (d) \(K\) is closed and bounded only if \(K\) is compact.
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