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Chapter 5: Problem 3

Which of the following sets are equal? $$ \begin{aligned} &A=\\{0,1,2\\} \\ &B=\\{x \in \mathbf{R} \mid-1 \leq x<3\\} \\ &C=\\{x \in \mathbf{R} \mid-1

Short Answer

Expert verified
The sets A and D are equal, as both contain the elements \(\\{0, 1, 2\\}\). Sets B, C, and E are not equal to any of the other sets.

Step by step solution

01

Analyze Set A

A = {0, 1, 2} is a finite set containing three integer elements, 0, 1, and 2.
02

Analyze Set B

B = {x 鈭� 鈩� | -1 鈮� x < 3} means the set contains all real numbers x such that -1 is less than or equal to x and x is less than 3. It includes all real numbers between -1 and 3, including -1 but excluding 3.
03

Analyze Set C

C = {x 鈭� 鈩� | -1 < x < 3} means the set contains all real numbers x such that -1 is less than x and x is less than 3. It includes all real numbers between -1 and 3, excluding both -1 and 3.
04

Analyze Set D

D = {x 鈭� 鈩� | -1 < x < 3} means the set contains all integer numbers x such that -1 is less than x and x is less than 3. It includes the integers 0, 1, and 2.
05

Analyze Set E

E = {x 鈭� 鈩も伜 | -1 < x < 3} means the set contains all positive integers x such that -1 is less than x and x is less than 3. It also includes the integers 1 and 2, but since the element 0 is not a positive integer, it is not included in this set.
06

Compare the sets

We can now compare the sets based on their elements: - Set A contains the elements {0, 1, 2}. - Set B contains all real numbers between -1 and 3, so it is not equal to any of the other sets. - Set C contains all real numbers between -1 and 3, excluding both -1 and 3, and it is not equal to any of the other sets. - Set D contains the elements {0, 1, 2}, which is the same as set A. - Set E contains the elements {1, 2}, so it is not equal to any of the other sets.
07

Conclusion

The sets A and D are equal, as both contain the elements {0, 1, 2}. Sets B, C, and E are not equal to any of the other sets.

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