Problem 69 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the rules regarding the negation of statements involving quantifiers to write the negation of the following statements. Which is true, the original statement or its negation? (a) Every isosceles triangle is equilateral. (b) There is a real number that is not an integer. (c) Every natural number is less than or equal to its square.

Step-by-Step Solution

Answer

(a) Negation is true. (b) Original is true. (c) Original is true.

Step 1: Understanding the Negation of Statements

To negate statements with quantifiers like 'every' or 'there exists', we apply the following rules: the negation of 'every' is 'there exists' and the negation of 'there exists' is 'every'. By using these rules, we can identify the negation of each statement.

Step 2: Negating Statement (a)

The original statement (a) says 'Every isosceles triangle is equilateral'. Using the negation rules, the negation is: 'There exists an isosceles triangle that is not equilateral'. This negation is true because not all isosceles triangles are equilateral.

Step 3: Negating Statement (b)

The original statement (b) says 'There is a real number that is not an integer'. The negation is: 'Every real number is an integer'. This negation is false, as there are many real numbers (like 1.5, π) that are not integers.

Step 4: Negating Statement (c)

The original statement (c) says 'Every natural number is less than or equal to its square'. The negation is: 'There exists a natural number that is greater than its square'. The negation is false, as every natural number (starting from 1) is less than or equal to its square.

Key Concepts

Quantifiers

Quantifiers are essential in both mathematical logic and everyday reasoning.
They help us express statements that involve generalization or existence. In mathematical logic, we often encounter two main types of quantifiers: **"universal quantifiers"** and **"existential quantifiers"**.
- **Universal quantifiers** are represented by the symbol \( \forall \), which translates to "for all" or "every." When a statement includes a universal quantifier, it claims that something is true for every possible instance. For example, "Every isosceles triangle is equilateral" implies that without exception, all isosceles triangles are equilateral.- **Existential quantifiers** are denoted by the symbol \( \exists \), which means "there exists" or "at least one." Statements with existential quantifiers claim that there is at least one instance for which the statement holds true. For instance, "There is a real number that is not an integer" suggests that at least one real number exists which isn't an integer, like \( \pi \).
Knowing how to negate quantified statements is crucial. The negation of \( \forall x, P(x) \) becomes \( \exists x, eg P(x) \). Similarly, \( \exists x, P(x) \) negates to \( \forall x, eg P(x) \). This switch is vital in forming accurate logical conclusions.

Logical Reasoning

Logical reasoning helps us analyze and deduce information to form conclusions. In the context of negating statements, it plays a pivotal role. The goal is to use logic to understand the truth or falsity of statements based on the negation rules of quantifiers.
When tackling logical reasoning tasks:
- Start by clearly understanding the original statement. Determine if the statement is using a universal or existential quantifier. - Apply the negation rules accurately. Remember, the negation of "every" is "there exists", while the negation of "there exists" is "every".
For instance, consider the statement "Every natural number is less than or equal to its square."
Using logical reasoning and negation rules, we derive "There exists a natural number that is greater than its square." You then evaluate its truth; in this example, the original statement is true, and its negation is false.
Using logical reasoning improves our ability to assess arguments and deduce valid conclusions, both critical skills in mathematical logic and solving complex problems effectively.

Mathematical Logic

Mathematical logic provides the foundation for understanding and manipulating mathematical statements formally. It's an essential branch of mathematics that delves into principles behind reasoning, including the action of negating statements.
- Core areas include propositional logic, predicate logic, and set theory, all of which heavily utilize quantifiers and logical operators.- Propositional logic involves evaluating statements that can only be true or false, whereas predicate logic introduces quantifiers, allowing for more complex expressions.
For example, the exercise of negating the statement "Every isosceles triangle is equilateral" (\( \forall x, P(x) \)) involves recognizing it's a universal quantifier and applying mathematical logic to substitute it with an existential quantifier: "There exists an isosceles triangle that is not equilateral" (\( \exists x, eg P(x) \)).
Engaging with mathematical logic enhances our ability to think critically and precisely. It also develops skills necessary for proving theorems, solving equations, and understanding computational logics, making it a priceless tool across various domains, from pure mathematics to computer science and beyond.