Find solutions to your questions with the help of IDNLearn.com's expert community. Ask your questions and receive reliable, detailed answers from our dedicated community of experts.
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.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.
