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Chapter 1: Problem 3

Determine whether each of the following conditional statements is true or false. (a) If \(10<7,\) then \(3=4\). (c) If \(10<7,\) then \(3+5=8\). (b) If \(7<10,\) then \(3=4\). (d) If \(7<10,\) then \(3+5=8\).

Short Answer

Expert verified
(a) True (b) True (c) False (d) True

Step by step solution

01

(a) Analyze the given statement

The given statement is "If \(10<7,\) then \(3=4\)." We see that the statement P is "\(10<7\)", which is false, and the statement Q is "\(3=4\)", which is also false. Since P is false, the conditional statement is true.
02

(b) Analyze the given statement

The given statement is "If \(10<7,\) then \(3+5=8\)." We see that the statement P is "\(10<7\)", which is false, and the statement Q is "\(3+5=8\)", which is true. Since P is false, the conditional statement is true.
03

(c) Analyze the given statement

The given statement is "If \(7<10,\) then \(3=4\)." We see that the statement P is "\(7<10\)", which is true, and the statement Q is "\(3=4\)", which is false. Since P is true and Q is false, the conditional statement is false.
04

(d) Analyze the given statement

The given statement is "If \(7<10,\) then \(3+5=8\)." We see that the statement P is "\(7<10\)", which is true, and the statement Q is "\(3+5=8\)", which is true. Since both P and Q are true, the conditional statement is true. So the truth values of the given conditional statements are: (a) True (b) True (c) False (d) True

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Key Concepts

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

Understanding Truth Values in Conditional Statements
In logic, conditional statements are sentences expressing "If... then..." scenarios. These statements have two parts: the antecedent ("If" part) and the consequent ("then" part). Understanding the truth values of these statements is crucial.
When examining conditional statements, we use truth values representing the truthfulness of these components:
  • "True" if the claim is factual.
  • "False" if the claim is incorrect.
A conditional statement is considered true unless the antecedent is true while the consequent is false. Here are the scenarios:
  • Both the antecedent and consequent are true: Conditional statement is true.
  • The antecedent is false and consequent is false: Conditional statement is true.
  • The antecedent is false and the consequent is true: Conditional statement is true.
  • The antecedent is true, but the consequent is false: Conditional statement is false.
The Role of Logical Reasoning
Logical reasoning involves evaluating statements and determining their validity. In the context of conditional statements, it helps us derive logical conclusions based on given premises.
When faced with conditional statements, logical reasoning involves these steps:
  • Identify the antecedent and consequent within the statement.
  • Assess the truthfulness of both parts based on known information or calculations.
  • Apply rules of truth values to decide if the overall statement is true or false.
By applying logical reasoning, we dissect statements systematically to reach valid conclusions. This process enables us to verify or dispute everyday claims efficiently, improving critical thinking skills.
Dealing with a False Antecedent
When a conditional statement contains a false antecedent, logical reasoning works uniquely. A false antecedent means the "if" part of the statement is incorrect. Despite any outcome in the consequent, the entire conditional statement is still considered true.
Why does this happen?
  • Conditional statements presume the antecedent's truth to evaluate its outcome鈥� an assumption must hold if we start with a false premise.
  • When the premise is false, the conclusion doesn't impact the statement's overall truth because the relationship is not tested under real conditions.
This concept helps explain why statements such as "If 10 < 7, then 3 = 4" are considered true in logic. The initial false premise means we do not verify the logical linkage between antecedent and consequent under normal conditions.

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Most popular questions from this chapter

Is the following statement true or false? Justify your conclusion. If \(a\) and \(b\) are nonnegative real numbers and \(a+b=0,\) then \(a=0\). Either give a counterexample to show that it is false or outline a proof by completing a know-show table.

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Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, \(a, b,\) and \(c\) are real numbers. Theorem If \(f\) is a quadratic function of the form \(f(x)=a x^{2}+b x+c\) and \(a c<0,\) then the function \(f\) has two \(x\) -intercepts. Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g(x)=-8 x^{2}+5 x-2\) (b) \(h(x)=-\frac{1}{3} x^{2}+3 x\) (c) \(k(x)=8 x^{2}-5 x-7\) (d) \(j(x)=-\frac{71}{99} x^{2}+210\) (e) \(f(x)=-4 x^{2}-3 x+7\) (f) \(F(x)=-x^{4}+x^{3}+9\)

Are the following statements true or false? Justify your conclusions. (a) If \(a, b\) and \(c\) are integers, then \(a b+a c\) is an even integer. (b) If \(b\) and \(c\) are odd integers and \(a\) is an integer, then \(a b+a c\) is an even integer.
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