Answer :
Certainly! Let's analyze each subset requested for the given set [tex]\( A = \{2, 4, 6, 8, 10, 12\} \)[/tex]. We'll identify the elements that belong to each subset step by step.
### (a) [tex]\( Q \)[/tex] - Set of odd numbers in [tex]\( A \)[/tex]
Odd numbers are those numbers that are not divisible by 2. Let's check each element in [tex]\( A \)[/tex]:
- 2 is not odd
- 4 is not odd
- 6 is not odd
- 8 is not odd
- 10 is not odd
- 12 is not odd
Hence, there are no odd numbers in [tex]\( A \)[/tex]. Thus,
[tex]\[ Q = \emptyset \][/tex]
### (b) [tex]\( P \)[/tex] - Set of prime numbers in [tex]\( A \)[/tex]
Prime numbers are those numbers greater than 1 that are divisible only by 1 and themselves. Let's check each element in [tex]\( A \)[/tex]:
- 2 is prime (divisible by 1 and 2)
- 4 is not prime (divisible by 1, 2, and 4)
- 6 is not prime (divisible by 1, 2, 3, and 6)
- 8 is not prime (divisible by 1, 2, 4, and 8)
- 10 is not prime (divisible by 1, 2, 5, and 10)
- 12 is not prime (divisible by 1, 2, 3, 4, 6, and 12)
Hence, we only have one prime number:
[tex]\[ P = \{2\} \][/tex]
### (c) [tex]\( C \)[/tex] - Set of composite numbers in [tex]\( A \)[/tex]
Composite numbers are those numbers greater than 1 that are not prime (i.e., they have divisors other than 1 and themselves). Let's check each element in [tex]\( A \)[/tex]:
- 2 is not composite (it’s prime)
- 4 is composite
- 6 is composite
- 8 is composite
- 10 is composite
- 12 is composite
Therefore, the composite numbers in [tex]\( A \)[/tex] are:
[tex]\[ C = \{4, 6, 8, 10, 12\} \][/tex]
### (d) [tex]\( F_{12} \)[/tex] - Set of factors of 12 in [tex]\( A \)[/tex]
Factors of 12 are the numbers that divide 12 without leaving a remainder. The factors of 12 are 1, 2, 3, 4, and 6. Let's check which of these factors are in [tex]\( A \)[/tex]:
- 2 is a factor of 12
- 4 is a factor of 12
- 6 is a factor of 12
- 8 is not a factor of 12
- 10 is not a factor of 12
- 12 is a factor of 12
Hence, the factors of 12 in [tex]\( A \)[/tex] are:
[tex]\[ F_{12} = \{2, 4, 6, 12\} \][/tex]
### (e) [tex]\( M_2 \)[/tex] - Set of multiples of 2 in [tex]\( A \)[/tex]
Multiples of 2 are those numbers that are divisible by 2. Let's identify the multiples of 2 in [tex]\( A \)[/tex]:
- 2 is a multiple of 2
- 4 is a multiple of 2
- 6 is a multiple of 2
- 8 is a multiple of 2
- 10 is a multiple of 2
- 12 is a multiple of 2
Therefore, all numbers in [tex]\( A \)[/tex] are multiples of 2:
[tex]\[ M_2 = \{2, 4, 6, 8, 10, 12\} \][/tex]
### (f) [tex]\( M_{15} \)[/tex] - Set of multiples of 15 in [tex]\( A \)[/tex]
Multiples of 15 are those numbers that are divisible by 15. Let's check if any element in [tex]\( A \)[/tex] is a multiple of 15:
- 2 is not a multiple of 15
- 4 is not a multiple of 15
- 6 is not a multiple of 15
- 8 is not a multiple of 15
- 10 is not a multiple of 15
- 12 is not a multiple of 15
Hence, there are no multiples of 15 in [tex]\( A \)[/tex]:
[tex]\[ M_{15} = \emptyset \][/tex]
Summarizing the results:
[tex]\[ Q = \emptyset \][/tex]
[tex]\[ P = \{2\} \][/tex]
[tex]\[ C = \{4, 6, 8, 10, 12\} \][/tex]
[tex]\[ F_{12} = \{2, 4, 6, 12\} \][/tex]
[tex]\[ M_2 = \{2, 4, 6, 8, 10, 12\} \][/tex]
[tex]\[ M_{15} = \emptyset \][/tex]
### (a) [tex]\( Q \)[/tex] - Set of odd numbers in [tex]\( A \)[/tex]
Odd numbers are those numbers that are not divisible by 2. Let's check each element in [tex]\( A \)[/tex]:
- 2 is not odd
- 4 is not odd
- 6 is not odd
- 8 is not odd
- 10 is not odd
- 12 is not odd
Hence, there are no odd numbers in [tex]\( A \)[/tex]. Thus,
[tex]\[ Q = \emptyset \][/tex]
### (b) [tex]\( P \)[/tex] - Set of prime numbers in [tex]\( A \)[/tex]
Prime numbers are those numbers greater than 1 that are divisible only by 1 and themselves. Let's check each element in [tex]\( A \)[/tex]:
- 2 is prime (divisible by 1 and 2)
- 4 is not prime (divisible by 1, 2, and 4)
- 6 is not prime (divisible by 1, 2, 3, and 6)
- 8 is not prime (divisible by 1, 2, 4, and 8)
- 10 is not prime (divisible by 1, 2, 5, and 10)
- 12 is not prime (divisible by 1, 2, 3, 4, 6, and 12)
Hence, we only have one prime number:
[tex]\[ P = \{2\} \][/tex]
### (c) [tex]\( C \)[/tex] - Set of composite numbers in [tex]\( A \)[/tex]
Composite numbers are those numbers greater than 1 that are not prime (i.e., they have divisors other than 1 and themselves). Let's check each element in [tex]\( A \)[/tex]:
- 2 is not composite (it’s prime)
- 4 is composite
- 6 is composite
- 8 is composite
- 10 is composite
- 12 is composite
Therefore, the composite numbers in [tex]\( A \)[/tex] are:
[tex]\[ C = \{4, 6, 8, 10, 12\} \][/tex]
### (d) [tex]\( F_{12} \)[/tex] - Set of factors of 12 in [tex]\( A \)[/tex]
Factors of 12 are the numbers that divide 12 without leaving a remainder. The factors of 12 are 1, 2, 3, 4, and 6. Let's check which of these factors are in [tex]\( A \)[/tex]:
- 2 is a factor of 12
- 4 is a factor of 12
- 6 is a factor of 12
- 8 is not a factor of 12
- 10 is not a factor of 12
- 12 is a factor of 12
Hence, the factors of 12 in [tex]\( A \)[/tex] are:
[tex]\[ F_{12} = \{2, 4, 6, 12\} \][/tex]
### (e) [tex]\( M_2 \)[/tex] - Set of multiples of 2 in [tex]\( A \)[/tex]
Multiples of 2 are those numbers that are divisible by 2. Let's identify the multiples of 2 in [tex]\( A \)[/tex]:
- 2 is a multiple of 2
- 4 is a multiple of 2
- 6 is a multiple of 2
- 8 is a multiple of 2
- 10 is a multiple of 2
- 12 is a multiple of 2
Therefore, all numbers in [tex]\( A \)[/tex] are multiples of 2:
[tex]\[ M_2 = \{2, 4, 6, 8, 10, 12\} \][/tex]
### (f) [tex]\( M_{15} \)[/tex] - Set of multiples of 15 in [tex]\( A \)[/tex]
Multiples of 15 are those numbers that are divisible by 15. Let's check if any element in [tex]\( A \)[/tex] is a multiple of 15:
- 2 is not a multiple of 15
- 4 is not a multiple of 15
- 6 is not a multiple of 15
- 8 is not a multiple of 15
- 10 is not a multiple of 15
- 12 is not a multiple of 15
Hence, there are no multiples of 15 in [tex]\( A \)[/tex]:
[tex]\[ M_{15} = \emptyset \][/tex]
Summarizing the results:
[tex]\[ Q = \emptyset \][/tex]
[tex]\[ P = \{2\} \][/tex]
[tex]\[ C = \{4, 6, 8, 10, 12\} \][/tex]
[tex]\[ F_{12} = \{2, 4, 6, 12\} \][/tex]
[tex]\[ M_2 = \{2, 4, 6, 8, 10, 12\} \][/tex]
[tex]\[ M_{15} = \emptyset \][/tex]
