The elements of a set may be sets themselves, as in {1, {1, 3}, {2}, 4}. Explain why the set {0} is not the same set as {{0}}.
Choose the correct answer below.
A. The set {0} is the set containing no elements, while {{0}} is a set containing a set 0.
B. The set {0} is the set containing the empty set and the empty set is a set with no elements, while {{0}} is a set containing a set 0.
C. The set {0} is the set containing the element, while {{0}} is a set containing the element 0.
D. The set {0} is the set containing the empty set and the empty set is a set with no elements, while {{0}} is a set containing the element 0.
Answer (2)
The set {0} contains the element 0.
The set {{0}} contains the set {0}.
Option A, B, and D are incorrect because they misinterpret the contents of {0}.
Option C is the closest to correct, although imperfectly worded, making it the best choice: OC .
Explanation
Understanding the Problem The question asks us to explain why the set {0} is not the same as the set {{0}} and to choose the correct explanation from the given options. The key is to understand that a set can contain other sets as elements.
Analyzing the Options Let's analyze each option:
Option A: The set {0} is the set containing no elements, while {{0}} is a set containing a set 0. This is incorrect because {0} contains the element 0, not no elements.
Option B: The set {0} is the set containing the empty set and the empty set is a set with no elements, while {{0}} is a set containing a set 0. This is incorrect because {0} contains the element 0, not the empty set.
Option C: The set {0} is the set containing the element, while {{0}} is a set containing the element 0. This is incorrect because {0} is the set containing the element 0, while {{0}} is the set containing the set {0}.
Option D: The set {0} is the set containing the empty set and the empty set is a set with no elements, while {{0}} is a set containing the element 0. This is incorrect because {0} contains the element 0, not the empty set.
Correct Understanding of the Sets Let's re-evaluate the options based on the correct understanding of the sets:
{0} is a set containing the element 0.
{{0}} is a set containing a set, where that set contains the element 0. In other words, {{0}} is a set containing the set {0}.
Choosing the Best Answer Based on the above analysis, none of the provided options accurately describe the difference between {0} and {{0}}. However, option C is closest, but it is poorly worded. The set {0} contains the element 0, while the set {{0}} contains the set {0}. Therefore, the best (though still imperfect) answer is OC .
Examples
In computer science, understanding the distinction between a set containing an element and a set containing another set is crucial. For example, a list of integers [1, 2, 3] is different from a list containing another list [[1, 2, 3]]. The first contains integers, while the second contains a list of integers. This distinction is important in data structures and algorithms.
The set {0} contains the element 0, whereas the set {{0}} contains the set {0} as its only element. Therefore, they are not the same due to their different compositions. The best answer choice is Option C.
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