Question

give an example of two sets that meet the following condition. if the condition is impossible to satisfy, explain why. the two sets are equivalent but not equal. choose the correct answer below. a. {x| x∈n and x ≤ 5} and {2,4,6,8} b. {x| x∈n and x < 5} and {1,2,3,4} c. {x| x∈n and x ≤ 5} and {1,2,3,4} d. {x| x∈n and x < 5} and {2,4,6,8} e. the condition is impossible to satisfy because sets that are equivalent are always equal

Explanation:

Step1: Define equivalent/equal sets

Equivalent sets have the same cardinality (number of elements); equal sets have identical elements.

Step2: Analyze Option A

First set: $\{x | x\in\mathbb{N} \text{ and } x\leq5\} = \{1,2,3,4,5\}$, cardinality = 5. Second set: $\{2,4,6,8\}$, cardinality = 4. Not equivalent.

Step3: Analyze Option B

First set: $\{x | x\in\mathbb{N} \text{ and } x<5\} = \{1,2,3,4\}$. Second set: $\{1,2,3,4\}$. Equal (and equivalent), does not meet the condition.

Step4: Analyze Option C

First set: $\{x | x\in\mathbb{N} \text{ and } x\leq5\} = \{1,2,3,4,5\}$, cardinality = 5. Second set: $\{1,2,3,4\}$, cardinality = 4. Not equivalent.

Step5: Analyze Option D

First set: $\{x | x\in\mathbb{N} \text{ and } x<5\} = \{1,2,3,4\}$, cardinality = 4. Second set: $\{2,4,6,8\}$, cardinality = 4. Equivalent (same size), not equal (different elements). Meets the condition.

Step6: Analyze Option E

False: Equivalent sets only need same cardinality, not identical elements.

Answer:

D. $\{x| x\in\mathbb{N} \text{ and } x < 5\}$ and $\{2,4,6,8\}$