To describe the set {2, 4, 6, 8, ...} using set-builder notation, we need to analyze the properties of the elements within this set.
1. The set consists of the numbers 2, 4, 6, 8, etc.
2. Each number in the set is a positive integer.
3. These numbers are all multiples of 2 and start from 2 onwards (i.e., it doesn’t include 0).
4. Since they are positive integers, we do not include negative multiples or zero.
Let's evaluate each of the provided options:
Option A: {x | x is a multiple of 2}
- This describes all multiples of 2, which would include negative numbers (-2, -4, etc.) and zero. Thus, it is not fully accurate for the given set.
Option B: {x | x is a non-negative multiple of 2}
- This describes all non-negative multiples of 2, which would include zero (0, 2, 4, 6, ...). The set we are concerned with does not include zero, so this is also not accurate.
Option C: {x | x is a multiple of a natural number and -2}
- This suggests numbers are multiples of both a natural number and -2, which includes negative values. This option doesn't make sense in standard mathematical contexts.
Option D: {x | x is a positive multiple of 2}
- This correctly describes our set: all x are positive multiples of 2. It excludes zero and negative multiples, perfectly matching the set {2, 4, 6, 8, ...}.
Therefore, the correct answer is:
OD. {x | x is a positive multiple of 2}
