Consider the following statements : 1. The set of all irrational numbers between \(\sqrt{2}\) and \(\sqrt{5}\) is an infinite set. 2. The set of all odd integers less than 100 is a finite set. Which of the statements given above is/are correct?
1 only
Understanding Finite and Infinite Sets
This question asks us to determine whether two different sets are finite or infinite. A finite set is a set with a limited number of elements, meaning we can count them and the counting process would end. An infinite set, on the other hand, has an unlimited number of elements; we could never finish counting them.
Analyzing Statement 1: Irrational Numbers Between \( \sqrt{2} \) and \( \sqrt{5} \)
Statement 1 considers the set of all irrational numbers between \( \sqrt{2} \) and \( \sqrt{5} \). Let's think about this interval on the number line. The value of \( \sqrt{2} \) is approximately 1.414, and the value of \( \sqrt{5} \) is approximately 2.236.
- The set is defined by the interval \( (\sqrt{2}, \sqrt{5}) \).
- We are looking specifically at irrational numbers within this interval.
- Irrational numbers are numbers that cannot be expressed as a simple fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Examples include \( \sqrt{2}, \sqrt{3}, \pi, e \).
A fundamental property of real numbers is that between any two distinct real numbers, there are infinitely many irrational numbers (and also infinitely many rational numbers). Since \( \sqrt{2} \) and \( \sqrt{5} \) are distinct real numbers (\( \sqrt{2} \neq \sqrt{5} \)), the interval between them, \( (\sqrt{2}, \sqrt{5}) \), contains infinitely many real numbers. Among these real numbers, there is an infinite supply of irrational numbers.
Therefore, the set of all irrational numbers between \( \sqrt{2} \) and \( \sqrt{5} \) is an infinite set.
Conclusion for Statement 1: Statement 1 is correct.
Analyzing Statement 2: Odd Integers Less Than 100
Statement 2 considers the set of all odd integers less than 100. Integers are whole numbers (positive, negative, or zero): ..., -3, -2, -1, 0, 1, 2, 3, ... Odd integers are integers that are not divisible by 2: ..., -5, -3, -1, 1, 3, 5, ...
We are looking for odd integers that are strictly less than 100. Let's list some of these odd integers:
- Positive odd integers less than 100: 1, 3, 5, ..., 97, 99.
- Zero is not an odd integer.
- Negative odd integers: -1, -3, -5, -7, ..., and so on.
The set of all odd integers less than 100 includes all positive odd integers up to 99 (1, 3, ..., 99) and all negative odd integers (..., -5, -3, -1). While the positive part (1, 3, ..., 99) is finite (there are \( \frac{99-1}{2} + 1 = \frac{98}{2} + 1 = 49 + 1 = 50 \) positive odd integers), the set of negative odd integers (..., -5, -3, -1) goes on forever towards negative infinity. For example, -101 is an odd integer less than 100, -103 is an odd integer less than 100, and this continues indefinitely.
Since the set of all odd integers less than 100 includes all negative odd integers, it has infinitely many elements.
Therefore, the set of all odd integers less than 100 is an infinite set.
Conclusion for Statement 2: Statement 2 is incorrect.
Summary of Analysis
- Statement 1: The set of all irrational numbers between \( \sqrt{2} \) and \( \sqrt{5} \) is an infinite set. (Correct)
- Statement 2: The set of all odd integers less than 100 is a finite set. (Incorrect)
Based on this analysis, only Statement 1 is correct.
| Statement | Set Description | Finite or Infinite? | Correctness |
|---|---|---|---|
| 1 | Irrational numbers between \( \sqrt{2} \) and \( \sqrt{5} \) | Infinite | Correct |
| 2 | Odd integers less than 100 | Infinite | Incorrect |
Therefore, only statement 1 is correct.
Revision Table: Set Properties
| Concept | Definition | Example |
|---|---|---|
| Finite Set | A set whose elements can be counted, ending with a non-negative integer. | {1, 2, 3}, Set of days in a week. |
| Infinite Set | A set whose elements cannot be counted in a finite amount of time. | Set of natural numbers {1, 2, 3, ...}, Set of points on a line. |
| Integer | A whole number (positive, negative, or zero). | ..., -2, -1, 0, 1, 2, ... |
| Odd Integer | An integer that is not divisible by 2. | ..., -3, -1, 1, 3, ... |
| Irrational Number | A real number that cannot be expressed as a simple fraction \( \frac{p}{q} \). | \( \sqrt{2} \), \( \pi \), \( e \) |
| Interval \( (a, b) \) | The set of all real numbers \( x \) such that \( a < x < b \). | Numbers between 0 and 1, excluding 0 and 1. |
Additional Information: Density of Numbers
The analysis of Statement 1 relies on the concept of the density of irrational numbers (and rational numbers) within the real number line. This means that between any two distinct real numbers, no matter how close they are, there are infinitely many numbers of both types (rational and irrational). This property makes the set of real numbers a continuum and fundamentally different from sets like integers.
For Statement 2, understanding the definition of "integers less than 100" is key. It includes all negative integers that are odd, which form an infinite sequence. If the question had specified "positive odd integers less than 100" or "natural numbers less than 100 that are odd", the set would have been finite.
Identifying whether a set is finite or infinite requires carefully considering the definition of the set and the properties of the numbers it contains.
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