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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.
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