From the given sets, which is an infinite set:
1. {x: x $\in$ N and (x-1)(x-2) = 0}
2. {x: x $\in$ N and x is prime number and less than 199}
3. {x: x $\in$ N and x$^5$ - 1 = 0}
4. {x: x $\in$ N and x is odd}

Question

Prepp Team · Accepted Answer

Identifying Infinite Sets from Given Natural Number Examples

The question asks us to identify which of the provided sets is an infinite set. A set is considered infinite if it contains an unlimited number of elements. Conversely, a finite set has a limited, countable number of elements.

Understanding Set Definitions

Let's analyze each set based on its definition:

Set 1 Analysis: Finite Set Example

The first set is defined as: {x: x $\in$ N and (x-1)(x-2) = 0}

Here, 'N' represents the set of natural numbers {1, 2, 3, ...}.
The condition is that the elements 'x' must be natural numbers satisfying the equation $(x-1)(x-2) = 0$.
Solving the equation:
- Either $x-1 = 0$, which gives $x = 1$.
- Or $x-2 = 0$, which gives $x = 2$.
Both $x = 1$ and $x = 2$ are natural numbers.
Therefore, this set is {1, 2}.
Since the set contains only two elements, it is a finite set.

Set 2 Analysis: Finite Set Example

The second set is defined as: {x: x $\in$ N and x is prime number and less than 199}

This set includes all natural numbers 'x' that are prime and also smaller than 199.
Examples of prime numbers are 2, 3, 5, 7, 11, etc.
The set consists of primes like {2, 3, 5, ..., p}, where 'p' is the largest prime number less than 199.
It is a known mathematical fact that there is a finite number of prime numbers less than any given integer.
Therefore, this set is a finite set.

Set 3 Analysis: Finite Set Example

The third set is defined as: {x: x $\in$ N and x⁵ - 1 = 0}

This set contains natural numbers 'x' that satisfy the equation $x⁵ - 1 = 0$.
Rearranging the equation gives $x⁵ = 1$.
The only natural number solution to this equation is $x = 1$ (since $1⁵ = 1$).
Therefore, this set is {1}.
Since the set contains only one element, it is a finite set.

Set 4 Analysis: Infinite Set Example

The fourth set is defined as: {x: x $\in$ N and x is odd}

This set includes all natural numbers 'x' that are odd.
Odd natural numbers are numbers that cannot be divided evenly by 2.
The sequence of odd natural numbers starts as 1, 3, 5, 7, 9, 11, and so on.
There is no upper limit to how large an odd natural number can be. We can always find another odd number by adding 2 to the previous one.
The set can be written as {1, 3, 5, 7, ...}.
Since there is no end to the sequence of odd natural numbers, this set contains an unlimited number of elements.
Therefore, this set is an infinite set.

Conclusion

Based on the analysis, the set {x: x $\in$ N and x is odd} is the only infinite set among the given options.

Prepp Team · Answer

{x: x $\in$ N and (x-1)(x-2) = 0}

Prepp Team · Answer

{x: x $\in$ N and x is prime number and less than 199}

Prepp Team · Answer

{x: x $\in$ N and x$^5$ - 1 = 0}

List-1	List-II
(A) If X and Y are two sets such that n(X)= 17, n(Y)=23, n(X $\cup$ Y)=38, then n(X $\cap$ Y) is	(I) 20
(B)) If n(X) = 28,n(Y) = 32,n(X$\cap$Y) = 10, then n(X$\cup$Y) is	(II) 10
(C) If n(X) = 10, then n(7X) is	(III) 50
(D) If n(Y) = 20, then n($\frac{Y}{2}$) is	(IV) 2