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Chapter 2: Problem 101

In Exercises 101-104, give examples of two sets that meet the given conditions. If the conditions are impossible to satisfy, explain why. The two sets are equivalent but not equal.

Short Answer

Expert verified
The sets {1, 2, 3} and {4, 5, 6} are equivalent (since they both have three elements) but not equal (since they do not share any common elements).

Step by step solution

01

Understanding equivalence and equality of sets

Two sets are said to be equivalent if they have the same number of elements. It does not matter what those elements are. Thus, the set {1, 2, 3} is equivalent to the set {4, 5, 6} because they both have three members. On the other hand, two sets are equal if they have exactly the same members. The set {1, 2, 3} is equal to the set {3, 2, 1} and not equal to {1, 2}, for example.
02

Picking two equivalent, but not equal sets

Now, let's select two sets which fulfill the condition of being equivalent but not equal. For this, it is important to pick two sets that have the same number of elements but different elements. As an example, one possible solution would be the sets {1, 2, 3} and {4, 5, 6}. Both have three elements (so, they are equivalent), but none of their elements are the same (so, they are not equal).
03

Confirming the solution

To confirm the solution, let's revisit the definitions of equivalence and equality of sets. Both the sets {1, 2, 3} and {4, 5, 6} have exactly three elements, hence they are equivalent. None of the elements of set {1, 2, 3} are in set {4, 5, 6}, hence they are not equal. Therefore, these two sets satisfy the given conditions in the exercise.

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