Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Let [tex]$A = \{2, 4, 6, 8, 10, 12\}$[/tex] be the given set. List the following subsets of [tex]$A$[/tex]:

(a) [tex][tex]$Q$[/tex][/tex] = Set of odd numbers

(b) [tex]$P$[/tex] = Set of prime numbers

(c) [tex]$C$[/tex] = Set of composite numbers

(d) [tex][tex]$F_{12}$[/tex][/tex] = Set of factors of 12

(e) [tex]$M_2$[/tex] = Set of multiples of 2

(f) [tex]$M_{15}$[/tex] = Set of multiples of 15


Sagot :

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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.