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}
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.
Important Questions from Set Theory and Venn Diagram (Notes)
A student is free to choose only Chemistry, only Biology or both. If out of $32$ students, Chemistry has been chosen by $16$ and Biology by $25$, then how many students have chosen Biology but not Chemistry?
- Fourteen of the students in a class are girls. Eight students in the class wear blue shirts. Two are neither girls nor wear blue shirts. Five students who wear blue shirts are girls. How many students are there in the class?
- In a group of 44 players, 26 play hockey, 24 play football and 24 play cricket. Eight of them play both hockey and football, 12 play both football and cricket, and 5 play all the three games. How many play both hockey and cricket?
- In a group of students, 30% play only cricket, 20% play only football and 10% play only basketball. 20% of the students play both football and cricket, 15% play both basketball and cricket, 10% play both football and basketball. 15 students play no games, while 5% of the students play all three games. What is the total number of students?
- Which of the following statements are true ?
A. Set A = {x : x $\in$ R and 2 < x < 3} is a null set.
B. Set A = {x : x $\in$ R and 2 < x < 4} is a singleton set.
C. Set A = {x : x $\in$ R and 1 < x < 9} is an infinite set.
D. Set A = {x : x $\in$ R and 1 < x < 9} is a finite set.
E. Set A = {a, b, c, d, e} and B = {c, d, a, e, b} are equal sets.
Choose the correct answer from the options given below :
