1. **State the problem:** We are given four pairs of sets (a), (b), (c), and (d). For each pair, we need to determine the relationship between the sets using the options: - Equivalent but not equal - Equal but not equivalent - Both equivalent and equal - Neither equivalent nor equal 2. **Recall definitions:** - Two sets are **equal** if they have exactly the same elements. - Two sets are **equivalent** if they have the same cardinality (number of elements), regardless of the elements themselves. 3. **Analyze each pair:** **(a)** - $A = \{85, 86, 87\}$ - $B = \{56, 57, 58\}$ - Both have 3 elements, so they are equivalent. - Elements differ, so not equal. - Conclusion: Equivalent but not equal. **(b)** - $A$ is the set of even numbers greater than 1 and less than 9, so $A = \{2, 4, 6, 8\}$ - $B = \{2, 4, 6, 8\}$ - Both sets have the same elements. - Conclusion: Both equivalent and equal. **(c)** - $A$ is integers greater than 20 and less than 24, so $A = \{21, 22, 23\}$ - $B$ is integers greater than 20, so $B = \{21, 22, 23, 24, 25, \ldots\}$ (infinite) - $A$ has 3 elements, $B$ is infinite. - Not equal, not equivalent. - Conclusion: Neither equivalent nor equal. **(d)** - $A = \{f, h, j, m\}$ has 4 elements - $B = \{f, d, c\}$ has 3 elements - Different cardinalities, so not equivalent. - Different elements, so not equal. - Conclusion: Neither equivalent nor equal. 4. **Final answers:** - (a) Equivalent but not equal - (b) Both equivalent and equal - (c) Neither equivalent nor equal - (d) Neither equivalent nor equal