Problem 64 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{S}\) be a non-empty subset of \(\mathrm{R}\). 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 \(\mathrm{P} ?\) (a) There is no rational number \(x \in \mathrm{S}\) such than \(x \leq 0 .\) (b) Every rational number \(x \in \mathrm{S}\) satisfies \(x \leq 0\). (c) \(x \in \mathrm{S}\) and \(x \leq 0 \Rightarrow x\) is not rational. (d) There is a rational number \(x \in \mathrm{S}\) such that \(x \leq 0 .\)

Step-by-Step Solution

Answer

Option (b) is the negation of the statement.

Step 1: Understanding the Given Statement

The statement\(\,P\) is \(x \in S\) where \(x\) is a rational number, such that \(x > 0\). This means there exists at least one rational number in \(S\) that is greater than zero.

Step 2: Formulating the Negation

The negation of the statement 'There exists a rational number \(x \in S\) such that \(x > 0\)' is 'There does not exist any rational number \(x \in S\) such that \(x > 0\)', or equivalently, 'Every rational number \(x \in S\) satisfies \(x \leq 0\)'.

Step 3: Analyzing Option (b)

Option (b) states: 'Every rational number \(x \in S\) satisfies \(x \leq 0\)'. This statement correctly represents the negation by confirming that no rational number is greater than zero.

Step 4: Reviewing Other Options

Options (a), (c), and (d) do not correctly represent the negation of \(P\). They either make different assumptions or misrepresent the condition of \(x > 0\).

Step 5: Conclusion

The correct negation of the statement \(P\) is expressed in option (b).

Key Concepts

Negation of Statements

In logical reasoning, understanding the concept of negation is crucial. The negation of a statement essentially involves constructing a new statement that contradicts or denies the original claim. Let's consider the statement: "There is a rational number \(x\) in \(S\) such that \(x > 0\)." This means at least one rational number greater than zero exists in the set \(S\).
To negate this, we must assert that it is not the case that there exists such a rational number. In simpler terms, negation means expressing the opposite outcome. So for this instance, the negation would be: "Every rational number \(x\) in \(S\) satisfies \(x \leq 0\)."
When working with negations, remember:

• "There exists" changes to "For all" or "Every."
• "It is not the case that" usually precedes the negated statement.
• In mathematical terms, inequalities may flip (e.g., \(>\) becomes \(\leq\)).

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers, and \(q eq 0\). They include all integers, fractions, and finite decimals. Understanding the nature of rational numbers helps in analyzing statements and their negations, especially when dealing with inequalities.
Key points about rational numbers are:

• They can be positive, negative, or zero.
• Rational numbers can be properly ordered on a number line, providing a clear sense of greater, lesser, and equal.
• When solving logical problems involving these numbers, both their positive and negative properties can come into play.

For example, when a statement mentions a rational number \(x > 0\), it specifies that \(x\) is positive. Therefore, understanding rational numbers enables us to determine whether statements about them, such as inequalities, logically hold true.

Inequalities

Inequalities in mathematics express a relationship between two expressions stating that one is greater than, less than, or not equal to another. In logical statements, inequalities often help define conditions that must be met.
The inequality \(x > 0\) indicates that \(x\) is positive. When negating statements involving inequalities, you often reverse the inequality sign. For example:

• The negation of "\(x > 0\)" becomes "\(x \leq 0\)".
• "\(x < 5\)" negates to "\(x \geq 5\)".

Inequalities can be strict (like \(>\) or \(<\)) or inclusive (like \(\geq\) or \(\leq\)), and this affects how they are used and negated in logical reasoning.
Understanding these concepts aids in correctly interpreting and manipulating inequalities, essential for identifying valid negations or proving assumptions in logical arguments.