Problem 70 - 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 natural number is rational. (b) There is a circle whose area is larger than \(9 \pi\). (c) Every real number is larger than its square.

Step-by-Step Solution

Answer

Statements (a) and (b) are true, while the negation of (c) is true.

Step 1: Understanding Quantifier Rules

Quantifiers include "for all" (universal quantifier, \(\forall\)) and "there exists" (existential quantifier, \(\exists\)). To negate a statement, replace \(\forall\) with \(\exists\) and \(\exists\) with \(\forall\), and negate the predicate.

Step 2: Negation of Statement (a)

The original statement is "Every natural number is rational," which is \(\forall n \in \mathbb{N}, n \text{ is rational}\). Its negation is: "There exists a natural number that is not rational," written as \(\exists n \in \mathbb{N}, n \text{ is not rational}\). The original statement is true because all natural numbers (e.g., 1, 2, 3) are also rational.

Step 3: Negation of Statement (b)

The original statement is "There is a circle whose area is larger than \(9\pi\)," which is \(\exists c, \, \text{Area}(c) > 9\pi\). Negating it gives us: "For every circle, the area is not larger than \(9\pi\)," written as \(\forall c, \, \text{Area}(c) \leq 9\pi\). The original statement is true because you can always find a circle with radius large enough (e.g., radius > 3) to have area greater than \(9\pi\).

Step 4: Negation of Statement (c)

The original statement is "Every real number is larger than its square," written as \(\forall x \in \mathbb{R}, x > x^2\). Its negation is: "There exists a real number that is not larger than its square," or \(\exists x \in \mathbb{R}, x \leq x^2\). The negation is true since, for example, for \(x = 2\) or \(x = 0.5\), \(x \leq x^2\).

Key Concepts

Quantifiers

Quantifiers are crucial components in logic that help us express the scope of statements. They allow us to discuss the quantity of particular elements that meet a certain condition or property. The two primary quantifiers used in mathematics and logic are the universal quantifier and the existential quantifier.

Understanding quantifiers enables us to frame statements that cover all elements within a set or just one or more elements.
The usage of quantifiers is prevalent in mathematical logic, set theory, and predicate logic. When dealing with quantifiers:

• The phrase "all" or "every" indicates a universal quantifier.
• The phrase "there exists" corresponds to an existential quantifier.

Changing a statement by negating its quantifiers changes the meaning significantly, flipping its truth value. For example, understanding whether statements about numbers being rational or relationships between numbers and shapes require precise quantification.

Therefore, a solid grasp of quantifiers helps us navigate complex logical constructs.

Universal Quantifier

The universal quantifier is represented by the symbol \(\forall\) and is used to indicate that a particular predicate or condition applies to all members of a set. It's like saying, "For any element you pick from this set, the given property holds true."

For instance, the statement "Every natural number is rational" uses the universal quantifier. In logical notation, this is written as \(\forall n \in \mathbb{N}, n \text{ is rational}\). It means no matter what number you choose from the set of all natural numbers, it will always be rational.

Negating a universally quantified statement involves replacing "for all" (\(\forall\)) with "there exists" (\(\exists\)) and negating the predicate. So, if we negate "Every natural number is rational," we get "There exists a natural number that is not rational."

This process helps us discover new truths and verify the original statements' validity.

Existential Quantifier

The existential quantifier, symbolized by \(\exists\), expresses that at least one member of a set satisfies a certain condition. It characterizes statements requiring just one element in a set to make the statement true.

Consider the example "There is a circle whose area is larger than \(9\pi\)." This statement can be represented as \(\exists c, \text{Area}(c) > 9\pi\). Essentially, it claims that you can find at least one circle whose area exceeds \(9\pi\).

When negating existential statements, we replace \(\exists\) with \(\forall\) and negate the inside predicate. For example, "There is a circle whose area is larger than \(9\pi\)" negates to "For every circle, the area is not larger than \(9\pi\)."

This type of transformation is fundamental in testing hypotheses and proving or disproving mathematical theorems effectively.

Predicate Logic

Predicate logic extends propositional logic by incorporating quantifiers and predicates, making it robust for expressing statements about objects and their properties.

A predicate is essentially a function that returns true or false. It takes objects as inputs and provides a true or false outcome based on whether those objects satisfy a certain condition.
In predicate logic, quantifiers \(\forall\) and \(\exists\) are used to communicate whether the properties represented by predicates apply universally or existentially.

• A universal quantifier asserts the truth of a predicate for all elements.
• An existential quantifier asserts the existence of at least one element for which the predicate is true.

Predicate logic proves invaluable in mathematics since it allows structuring precise arguments about infinite sets, like the set of all real numbers or all natural numbers. Mastering predicate logic is key to decoding statements rigorously and logically.