Question

e. which of the following describes the set {x | x > x + 2 and x ∈ n}? a. it is not well defined. there is no clear meaning of the description x > x + 2. b. it is well defined. it is the set of all whole numbers, w. c. it is well defined. it is the set of all natural numbers, n. d. it is not well defined. there is no clear meaning of the description x ∈ n. e. it is not well defined. given a natural number x, it is not possible to tell whether x > x + 2 is true or false. f. it is well defined. it is the empty set ∅.

Explanation:

Step1: Analyze the inequality \( x > x + 2 \)

Subtract \( x \) from both sides: \( x - x > x + 2 - x \), which simplifies to \( 0 > 2 \).

Step2: Determine the set

Since \( 0 > 2 \) is a false statement, there are no natural numbers \( x \) that satisfy \( x > x + 2 \). So the set is the empty set.

Step3: Evaluate the options

• Option A: Says "It is not well defined...", but the inequality is clear (just false for all \( x\in\mathbb{N} \)), so A is wrong.
• Option B: Claims it's the set of whole numbers, but we saw no solutions, so B is wrong.
• Option C: Claims it's the set of natural numbers, but no solutions, so C is wrong.
• Option D: Says "It is not well defined...", but the description is clear, so D is wrong.
• Option E: Says "It is not well defined...", incorrect.
• Option F: Says "It is well defined. It is the empty set \( \varnothing \)", which matches our conclusion.

Answer:

F. It is well defined. It is the empty set \( \varnothing \)