Problem 30 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(S\) be a non-empty subset of \(R\). Consider the following statement: \(P\) : There is a rational number \(x \in S\) such that \(x>0\). Which of the following statements is the negation of the statement \(P\) ? (A) There is no rational number \(x \in S\) such that \(x \leq 0\) (B) Every rational number \(x \in S\) satisfies \(x \leq 0\) (C) \(x \in S\) and \(x \leq 0 \Rightarrow x\) is not rational (D) There is a rational number \(x \in S\) such that \(x \leq 0\)

Step-by-Step Solution

Answer

The negation of statement \(P\) is (B): Every rational number \(x \in S\) satisfies \(x \leq 0\).

Step 1: Understand the Original Statement

The original statement \(P\) asserts that there exists a rational number \(x\) in the subset \(S\) such that \(x > 0\). This means that among the elements in \(S\), at least one is a positive rational number.

Step 2: Formulate Negation of Statement

The negation of the existential statement "there exists" transforms it into a universal statement. So, negating \(P\) results in the statement that no rational number in \(S\) is greater than zero.

Step 3: Rewrite Negation in Proper Form

The negation, "no rational number in \(S\) is greater than 0," is equivalent to saying "every rational number \(x\) in \(S\) satisfies \(x \leq 0\)." This reformulation captures the universal statement that any rational number in \(S\) is non-positive.

Step 4: Match with Given Options

Examine the options and find which one matches the derived negation. Option (B) states "Every rational number \(x \in S\) satisfies \(x \leq 0\)," which exactly matches the rewritten negation.

Step 5: Conclusion

Therefore, the negation of statement \(P\), which is "There is a rational number \(x \in S\) such that \(x > 0\)," is Option (B): "Every rational number \(x \in S\) satisfies \(x \leq 0\)."

Key Concepts

Existential Quantifier

The existential quantifier is a fundamental concept in logic and mathematics. It is used to express that there exists at least one element in a set that satisfies a given property. This is often denoted with the symbol \(\exists\). For example, if we say \(\exists x \in S\), it means "there is at least one \(x\) in the set \(S\)."

In logical statements, the existential quantifier helps pinpoint that at least one solution or case exists for a condition. When using it, we typically encounter it in statements like "there exists a rational number \(x\) such that \(x > 0\)." This indicates that within the set of numbers under consideration, at least one number meets the condition of being positive and rational.

Negating an existential statement involves converting it to a universal statement. So, when negating the statement "there exists a rational number \(x\) such that \(x > 0\)," it becomes "no rational number \(x\) is greater than 0," or "every rational number \(x\) satisfies \(x \leq 0\)." This transition from existential to universal is essential in creating the opposite of the initial statement.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, with the denominator not equal to zero. Typically, a rational number \(x\) is written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\).

Rational numbers include positive numbers, negative numbers, and zero. They are dense in the real number line, meaning between any two numbers, you can find a rational number.

• Positive rational numbers: Greater than zero, like \(\frac{2}{3}\) or \(5\).
• Negative rational numbers: Less than zero, like \(-\frac{4}{7}\) or \(-3\).
• Zero: Considered rational since it can be represented as \(\frac{0}{1}\).

The property of rational numbers being able to be expressed exactly as fractions distinguishes them from irrational numbers, which cannot. Understanding this helps identify them within a set, especially when concepts like existential quantifiers are used to determine if such numbers fulfill certain properties.

Universal Quantifier

The universal quantifier is another crucial concept in logic. It is represented by the symbol \(\forall\), and it implies that a property or condition applies to every element within a set.

For example, the statement \(\forall x \in S, x \leq 0\) means that every number \(x\) in the set \(S\) is less than or equal to zero. In logic, when you encounter the universal quantifier, it is about understanding that the property holds across the entire set without exceptions.

The significance of the universal quantifier becomes evident when transforming existential statements to their negations. As covered before, the negation of an existential statement uses the universal quantifier. In our case, we moved from "there is a rational number \(x\) such that \(x > 0\)" to "for all rational numbers \(x\), \(x \leq 0\)."

This switch is pivotal in logical reasoning and proofs, as it allows us to thoroughly express and explore all circumstances within the defined set or scope calmly and comprehensively.