Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine if the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are equal, we need to understand their elements and verify if both sets contain the exact same elements. Here is the step-by-step solution:
### Set [tex]\( A \)[/tex]
The elements of set [tex]\( A \)[/tex] are defined by:
[tex]\[ A = \{ a^3 + 2 \mid a \in \{ 1, 2, \ldots, 20 \} \} \][/tex]
This means that for each integer [tex]\( a \)[/tex] from 1 to 20, we calculate [tex]\( a^3 + 2 \)[/tex]. Let's compute these values:
1. For [tex]\( a = 1 \)[/tex]: [tex]\( 1^3 + 2 = 1 + 2 = 3 \)[/tex]
2. For [tex]\( a = 2 \)[/tex]: [tex]\( 2^3 + 2 = 8 + 2 = 10 \)[/tex]
3. For [tex]\( a = 3 \)[/tex]: [tex]\( 3^3 + 2 = 27 + 2 = 29 \)[/tex]
4. For [tex]\( a = 4 \)[/tex]: [tex]\( 4^3 + 2 = 64 + 2 = 66 \)[/tex]
5. For [tex]\( a = 5 \)[/tex]: [tex]\( 5^3 + 2 = 125 + 2 = 127 \)[/tex]
6. For [tex]\( a = 6 \)[/tex]: [tex]\( 6^3 + 2 = 216 + 2 = 218 \)[/tex]
7. For [tex]\( a = 7 \)[/tex]: [tex]\( 7^3 + 2 = 343 + 2 = 345 \)[/tex]
8. For [tex]\( a = 8 \)[/tex]: [tex]\( 8^3 + 2 = 512 + 2 = 514 \)[/tex]
9. For [tex]\( a = 9 \)[/tex]: [tex]\( 9^3 + 2 = 729 + 2 = 731 \)[/tex]
10. For [tex]\( a = 10 \)[/tex]: [tex]\( 10^3 + 2 = 1000 + 2 = 1002 \)[/tex]
11. For [tex]\( a = 11 \)[/tex]: [tex]\( 11^3 + 2 = 1331 + 2 = 1333 \)[/tex]
12. For [tex]\( a = 12 \)[/tex]: [tex]\( 12^3 + 2 = 1728 + 2 = 1730 \)[/tex]
13. For [tex]\( a = 13 \)[/tex]: [tex]\( 13^3 + 2 = 2197 + 2 = 2199 \)[/tex]
14. For [tex]\( a = 14 \)[/tex]: [tex]\( 14^3 + 2 = 2744 + 2 = 2746 \)[/tex]
15. For [tex]\( a = 15 \)[/tex]: [tex]\( 15^3 + 2 = 3375 + 2 = 3377 \)[/tex]
16. For [tex]\( a = 16 \)[/tex]: [tex]\( 16^3 + 2 = 4096 + 2 = 4098 \)[/tex]
17. For [tex]\( a = 17 \)[/tex]: [tex]\( 17^3 + 2 = 4913 + 2 = 4915 \)[/tex]
18. For [tex]\( a = 18 \)[/tex]: [tex]\( 18^3 + 2 = 5832 + 2 = 5834 \)[/tex]
19. For [tex]\( a = 19 \)[/tex]: [tex]\( 19^3 + 2 = 6859 + 2 = 6861 \)[/tex]
20. For [tex]\( a = 20 \)[/tex]: [tex]\( 20^3 + 2 = 8000 + 2 = 8002 \)[/tex]
Thus, set [tex]\( A \)[/tex] is:
[tex]\[ A = \{ 3, 10, 29, 66, 127, 218, 345, 514, 731, 1002, 1333, 1730, 2199, 2746, 3377, 4098, 4915, 5834, 6861, 8002 \} \][/tex]
### Set [tex]\( B \)[/tex]
The elements of set [tex]\( B \)[/tex] are defined by:
[tex]\[ B = \{ 29 \} \cup \{ b^5 - 4 \mid b \in \{ 1, 2, \ldots, 12 \} \} \][/tex]
This means we have the fixed element 29, and for each integer [tex]\( b \)[/tex] from 1 to 12, we calculate [tex]\( b^5 - 4 \)[/tex]. Let's compute these values:
1. [tex]\( 29 \)[/tex]
2. For [tex]\( b = 1 \)[/tex]: [tex]\( 1^5 - 4 = 1 - 4 = -3 \)[/tex]
3. For [tex]\( b = 2 \)[/tex]: [tex]\( 2^5 - 4 = 32 - 4 = 28 \)[/tex]
4. For [tex]\( b = 3 \)[/tex]: [tex]\( 3^5 - 4 = 243 - 4 = 239 \)[/tex]
5. For [tex]\( b = 4 \)[/tex]: [tex]\( 4^5 - 4 = 1024 - 4 = 1020 \)[/tex]
6. For [tex]\( b = 5 \)[/tex]: [tex]\( 5^5 - 4 = 3125 - 4 = 3121 \)[/tex]
7. For [tex]\( b = 6 \)[/tex]: [tex]\( 6^5 - 4 = 7776 - 4 = 7772 \)[/tex]
8. For [tex]\( b = 7 \)[/tex]: [tex]\( 7^5 - 4 = 16807 - 4 = 16803 \)[/tex]
9. For [tex]\( b = 8 \)[/tex]: [tex]\( 8^5 - 4 = 32768 - 4 = 32764 \)[/tex]
10. For [tex]\( b = 9 \)[/tex]: [tex]\( 9^5 - 4 = 59049 - 4 = 59045 \)[/tex]
11. For [tex]\( b = 10 \)[/tex]: [tex]\( 10^5 - 4 = 100000 - 4 = 99996 \)[/tex]
12. For [tex]\( b = 11 \)[/tex]: [tex]\( 11^5 - 4 = 161051 - 4 = 161047 \)[/tex]
13. For [tex]\( b = 12 \)[/tex]: [tex]\( 12^5 - 4 = 248832 - 4 = 248828 \)[/tex]
Thus, set [tex]\( B \)[/tex] is:
[tex]\[ B = \{ 29, -3, 28, 239, 1020, 3121, 7772, 16803, 32764, 59045, 99996, 161047, 248828 \} \][/tex]
### Comparison of Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
We now compare the elements of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Set [tex]\( A \)[/tex] is:
[tex]\[ \{ 3, 10, 29, 66, 127, 218, 345, 514, 731, 1002, 1333, 1730, 2199, 2746, 3377, 4098, 4915, 5834, 6861, 8002 \} \][/tex]
Set [tex]\( B \)[/tex] is:
[tex]\[ \{ 29, -3, 28, 239, 1020, 3121, 7772, 16803, 32764, 59045, 99996, 161047, 248828 \} \][/tex]
Comparing both sets, we can see they don't have the same elements. Therefore:
### Conclusion
The sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not equal.
### Set [tex]\( A \)[/tex]
The elements of set [tex]\( A \)[/tex] are defined by:
[tex]\[ A = \{ a^3 + 2 \mid a \in \{ 1, 2, \ldots, 20 \} \} \][/tex]
This means that for each integer [tex]\( a \)[/tex] from 1 to 20, we calculate [tex]\( a^3 + 2 \)[/tex]. Let's compute these values:
1. For [tex]\( a = 1 \)[/tex]: [tex]\( 1^3 + 2 = 1 + 2 = 3 \)[/tex]
2. For [tex]\( a = 2 \)[/tex]: [tex]\( 2^3 + 2 = 8 + 2 = 10 \)[/tex]
3. For [tex]\( a = 3 \)[/tex]: [tex]\( 3^3 + 2 = 27 + 2 = 29 \)[/tex]
4. For [tex]\( a = 4 \)[/tex]: [tex]\( 4^3 + 2 = 64 + 2 = 66 \)[/tex]
5. For [tex]\( a = 5 \)[/tex]: [tex]\( 5^3 + 2 = 125 + 2 = 127 \)[/tex]
6. For [tex]\( a = 6 \)[/tex]: [tex]\( 6^3 + 2 = 216 + 2 = 218 \)[/tex]
7. For [tex]\( a = 7 \)[/tex]: [tex]\( 7^3 + 2 = 343 + 2 = 345 \)[/tex]
8. For [tex]\( a = 8 \)[/tex]: [tex]\( 8^3 + 2 = 512 + 2 = 514 \)[/tex]
9. For [tex]\( a = 9 \)[/tex]: [tex]\( 9^3 + 2 = 729 + 2 = 731 \)[/tex]
10. For [tex]\( a = 10 \)[/tex]: [tex]\( 10^3 + 2 = 1000 + 2 = 1002 \)[/tex]
11. For [tex]\( a = 11 \)[/tex]: [tex]\( 11^3 + 2 = 1331 + 2 = 1333 \)[/tex]
12. For [tex]\( a = 12 \)[/tex]: [tex]\( 12^3 + 2 = 1728 + 2 = 1730 \)[/tex]
13. For [tex]\( a = 13 \)[/tex]: [tex]\( 13^3 + 2 = 2197 + 2 = 2199 \)[/tex]
14. For [tex]\( a = 14 \)[/tex]: [tex]\( 14^3 + 2 = 2744 + 2 = 2746 \)[/tex]
15. For [tex]\( a = 15 \)[/tex]: [tex]\( 15^3 + 2 = 3375 + 2 = 3377 \)[/tex]
16. For [tex]\( a = 16 \)[/tex]: [tex]\( 16^3 + 2 = 4096 + 2 = 4098 \)[/tex]
17. For [tex]\( a = 17 \)[/tex]: [tex]\( 17^3 + 2 = 4913 + 2 = 4915 \)[/tex]
18. For [tex]\( a = 18 \)[/tex]: [tex]\( 18^3 + 2 = 5832 + 2 = 5834 \)[/tex]
19. For [tex]\( a = 19 \)[/tex]: [tex]\( 19^3 + 2 = 6859 + 2 = 6861 \)[/tex]
20. For [tex]\( a = 20 \)[/tex]: [tex]\( 20^3 + 2 = 8000 + 2 = 8002 \)[/tex]
Thus, set [tex]\( A \)[/tex] is:
[tex]\[ A = \{ 3, 10, 29, 66, 127, 218, 345, 514, 731, 1002, 1333, 1730, 2199, 2746, 3377, 4098, 4915, 5834, 6861, 8002 \} \][/tex]
### Set [tex]\( B \)[/tex]
The elements of set [tex]\( B \)[/tex] are defined by:
[tex]\[ B = \{ 29 \} \cup \{ b^5 - 4 \mid b \in \{ 1, 2, \ldots, 12 \} \} \][/tex]
This means we have the fixed element 29, and for each integer [tex]\( b \)[/tex] from 1 to 12, we calculate [tex]\( b^5 - 4 \)[/tex]. Let's compute these values:
1. [tex]\( 29 \)[/tex]
2. For [tex]\( b = 1 \)[/tex]: [tex]\( 1^5 - 4 = 1 - 4 = -3 \)[/tex]
3. For [tex]\( b = 2 \)[/tex]: [tex]\( 2^5 - 4 = 32 - 4 = 28 \)[/tex]
4. For [tex]\( b = 3 \)[/tex]: [tex]\( 3^5 - 4 = 243 - 4 = 239 \)[/tex]
5. For [tex]\( b = 4 \)[/tex]: [tex]\( 4^5 - 4 = 1024 - 4 = 1020 \)[/tex]
6. For [tex]\( b = 5 \)[/tex]: [tex]\( 5^5 - 4 = 3125 - 4 = 3121 \)[/tex]
7. For [tex]\( b = 6 \)[/tex]: [tex]\( 6^5 - 4 = 7776 - 4 = 7772 \)[/tex]
8. For [tex]\( b = 7 \)[/tex]: [tex]\( 7^5 - 4 = 16807 - 4 = 16803 \)[/tex]
9. For [tex]\( b = 8 \)[/tex]: [tex]\( 8^5 - 4 = 32768 - 4 = 32764 \)[/tex]
10. For [tex]\( b = 9 \)[/tex]: [tex]\( 9^5 - 4 = 59049 - 4 = 59045 \)[/tex]
11. For [tex]\( b = 10 \)[/tex]: [tex]\( 10^5 - 4 = 100000 - 4 = 99996 \)[/tex]
12. For [tex]\( b = 11 \)[/tex]: [tex]\( 11^5 - 4 = 161051 - 4 = 161047 \)[/tex]
13. For [tex]\( b = 12 \)[/tex]: [tex]\( 12^5 - 4 = 248832 - 4 = 248828 \)[/tex]
Thus, set [tex]\( B \)[/tex] is:
[tex]\[ B = \{ 29, -3, 28, 239, 1020, 3121, 7772, 16803, 32764, 59045, 99996, 161047, 248828 \} \][/tex]
### Comparison of Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
We now compare the elements of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Set [tex]\( A \)[/tex] is:
[tex]\[ \{ 3, 10, 29, 66, 127, 218, 345, 514, 731, 1002, 1333, 1730, 2199, 2746, 3377, 4098, 4915, 5834, 6861, 8002 \} \][/tex]
Set [tex]\( B \)[/tex] is:
[tex]\[ \{ 29, -3, 28, 239, 1020, 3121, 7772, 16803, 32764, 59045, 99996, 161047, 248828 \} \][/tex]
Comparing both sets, we can see they don't have the same elements. Therefore:
### Conclusion
The sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not equal.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
