Answer :
To classify the given sets into empty sets, finite sets, and infinite sets, and mention the cardinal number for finite sets, we will analyze each set one by one:
(a) [tex]\( Z = \{ \text{prime numbers from 2 to 11} \} \)[/tex]:
- To determine the set [tex]\( Z \)[/tex], let's list the prime numbers between 2 and 11:
- Prime numbers in this range are: 2, 3, 5, 7, and 11.
- Therefore, the set [tex]\( Z = \{ 2, 3, 5, 7, 11 \} \)[/tex].
- This set is finite since it contains a specific list of numbers.
- The cardinal number (the number of elements) of set [tex]\( Z \)[/tex] is 5.
Hence, [tex]\( Z \)[/tex] is a finite set with a cardinal number of 5.
(b) [tex]\( P = \{ x \mid x \text{ is an even prime number} \} \)[/tex]:
- Checking for even prime numbers:
- The only even prime number is 2.
- Therefore, the set [tex]\( P = \{ 2 \} \)[/tex].
- This set is finite since it contains a specific list of numbers.
- The cardinal number of set [tex]\( P \)[/tex] is 1.
Hence, [tex]\( P \)[/tex] is a finite set with a cardinal number of 1.
(c) [tex]\( Q = \{ x \mid x \text{ is a quadrilateral having 5 sides} \} \)[/tex]:
- By definition, a quadrilateral is a polygon with exactly 4 sides.
- There cannot be a quadrilateral with 5 sides.
- Therefore, the set [tex]\( Q \)[/tex] has no elements.
Hence, [tex]\( Q \)[/tex] is an empty set.
(d) [tex]\( R = \{ x \mid x \in \mathbb{I}, -5 < x < 2 \} \)[/tex]:
- The set [tex]\( \mathbb{I} \)[/tex] refers to integers, and we need integers in the range [tex]\( -5 < x < 2 \)[/tex]:
- The integers between -5 and 2 are: -4, -3, -2, -1, 0, and 1.
- Therefore, the set [tex]\( R = \{ -4, -3, -2, -1, 0, 1 \} \)[/tex].
- This set is finite since it contains a specific list of numbers.
- The cardinal number of set [tex]\( R \)[/tex] is 6.
Hence, [tex]\( R \)[/tex] is a finite set with a cardinal number of 6.
(e) [tex]\( V = \{ x \mid x \text{ is a 2-digit number such that the sum of digits is 6} \} \)[/tex]:
- To find [tex]\( x \)[/tex], let's consider two-digit numbers where the sum of digits is 6:
- Examples include: 15 (1+5), 24 (2+4), 33 (3+3), 42 (4+2), 51 (5+1), 60 (6+0).
- Therefore, the set [tex]\( V = \{ 15, 24, 33, 42, 51, 60 \} \)[/tex].
- This set is finite since it contains a specific list of numbers.
- The cardinal number of set [tex]\( V \)[/tex] is 6.
Hence, [tex]\( V \)[/tex] is a finite set with a cardinal number of 6.
In summary:
- [tex]\( Z \)[/tex] is a finite set with a cardinal number of 5.
- [tex]\( P \)[/tex] is a finite set with a cardinal number of 1.
- [tex]\( Q \)[/tex] is an empty set.
- [tex]\( R \)[/tex] is a finite set with a cardinal number of 6.
- [tex]\( V \)[/tex] is a finite set with a cardinal number of 6.
(a) [tex]\( Z = \{ \text{prime numbers from 2 to 11} \} \)[/tex]:
- To determine the set [tex]\( Z \)[/tex], let's list the prime numbers between 2 and 11:
- Prime numbers in this range are: 2, 3, 5, 7, and 11.
- Therefore, the set [tex]\( Z = \{ 2, 3, 5, 7, 11 \} \)[/tex].
- This set is finite since it contains a specific list of numbers.
- The cardinal number (the number of elements) of set [tex]\( Z \)[/tex] is 5.
Hence, [tex]\( Z \)[/tex] is a finite set with a cardinal number of 5.
(b) [tex]\( P = \{ x \mid x \text{ is an even prime number} \} \)[/tex]:
- Checking for even prime numbers:
- The only even prime number is 2.
- Therefore, the set [tex]\( P = \{ 2 \} \)[/tex].
- This set is finite since it contains a specific list of numbers.
- The cardinal number of set [tex]\( P \)[/tex] is 1.
Hence, [tex]\( P \)[/tex] is a finite set with a cardinal number of 1.
(c) [tex]\( Q = \{ x \mid x \text{ is a quadrilateral having 5 sides} \} \)[/tex]:
- By definition, a quadrilateral is a polygon with exactly 4 sides.
- There cannot be a quadrilateral with 5 sides.
- Therefore, the set [tex]\( Q \)[/tex] has no elements.
Hence, [tex]\( Q \)[/tex] is an empty set.
(d) [tex]\( R = \{ x \mid x \in \mathbb{I}, -5 < x < 2 \} \)[/tex]:
- The set [tex]\( \mathbb{I} \)[/tex] refers to integers, and we need integers in the range [tex]\( -5 < x < 2 \)[/tex]:
- The integers between -5 and 2 are: -4, -3, -2, -1, 0, and 1.
- Therefore, the set [tex]\( R = \{ -4, -3, -2, -1, 0, 1 \} \)[/tex].
- This set is finite since it contains a specific list of numbers.
- The cardinal number of set [tex]\( R \)[/tex] is 6.
Hence, [tex]\( R \)[/tex] is a finite set with a cardinal number of 6.
(e) [tex]\( V = \{ x \mid x \text{ is a 2-digit number such that the sum of digits is 6} \} \)[/tex]:
- To find [tex]\( x \)[/tex], let's consider two-digit numbers where the sum of digits is 6:
- Examples include: 15 (1+5), 24 (2+4), 33 (3+3), 42 (4+2), 51 (5+1), 60 (6+0).
- Therefore, the set [tex]\( V = \{ 15, 24, 33, 42, 51, 60 \} \)[/tex].
- This set is finite since it contains a specific list of numbers.
- The cardinal number of set [tex]\( V \)[/tex] is 6.
Hence, [tex]\( V \)[/tex] is a finite set with a cardinal number of 6.
In summary:
- [tex]\( Z \)[/tex] is a finite set with a cardinal number of 5.
- [tex]\( P \)[/tex] is a finite set with a cardinal number of 1.
- [tex]\( Q \)[/tex] is an empty set.
- [tex]\( R \)[/tex] is a finite set with a cardinal number of 6.
- [tex]\( V \)[/tex] is a finite set with a cardinal number of 6.
Step-by-step explanation:
(a) Z = The set of prime numbers from 2 to 11
ans: {2,3,5,7,11}
finite set
(b) P = {x: x is an even prime number}
ans: {2}
finite set
(c) Q = { x : x is a quadrilateral having 5 sides}
ans: null set
empty set
(d) R = { x: x belongs to I -5< x < 2}
ans: empty set
(e) V = { x: x is a 2-digit number such that the sum of digits is 6 }
ans: {15, 24 ,33, 42, 51, 60}
finite set
