Problem 21 - Discrete Mathematics with Applications

Question

Mark each as true or false. $$ \emptyset \in\\{\varnothing\\} $$

Step-by-Step Solution

Answer

The statement \(\emptyset \in \{\emptyset\}\) is true, as the set containing the empty set \(\{\emptyset\}\) has one element, which is the empty set itself.

Step 1: Understanding the empty set and the set containing the empty set

The empty set, denoted by \(\emptyset\), is a set with no elements in it. On the other hand, the set containing the empty set, \(\{\emptyset\}\), has one element, which is the empty set itself.

Step 2: Comparing the two sets

Now let's determine if the given statement, \(\emptyset \in \{\emptyset\}\), is true or false. The statement is saying that the empty set is an element of the set containing the empty set. Recall that the set containing the empty set, \(\{\emptyset\}\), has one element, which is the empty set itself, i.e., \(\{\emptyset\} = \{ \emptyset \} \). As we can see, the empty set is indeed an element of the set containing the empty set.

Step 3: Conclusion

Therefore, the statement \(\emptyset \in \{\emptyset\}\) is true.

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with the collection of objects, known as elements or members. In study or application, sets are often denoted with curly braces, such as \( \{a, b, c\} \), and each element is listed inside the braces, separated by commas.

One of the most basic concepts in set theory is the empty set, symbolized by \( \emptyset \). It represents a set with no elements. It's important to distinguish the empty set from a set that contains the empty set, as the latter still has an element—\( \emptyset \) itself—though this element denotes 'nothingness'. Understanding these foundational concepts is crucial for further mathematical explorations, including functions, probability, and algebra.

Elements of a Set

In set theory, an element is a singular object contained within a set. The notation \( a \in A \) is used to denote that 'a' is an element of the set 'A'. Conversely, if 'a' is not in 'A', we write \( a otin A \).

An element can be anything: a number, a letter, a symbol, or even another set. This leads to the possibility of sets containing sets, known as nested sets. While the elements in a set are distinct, meaning no two elements can be the same, a set can have any number of elements, from zero (the empty set) to an infinite number. Understanding the unique identity of each element is essential for grasping more complex concepts like subsets, unions, intersections, and Cartesian products in set theory.

Subsets

A subset is a set whose elements are all contained within another set. If we have two sets, 'A' and 'B', we say that 'A' is a subset of 'B', denoted by \( A \subseteq B \), if every element of 'A' is also in 'B'. It's essential to note that any set is a subset of itself, and the empty set, being devoid of elements, is a subset of every set.

Special Consideration for the Empty Set

Considering the unique nature of the empty set, it may seem counterintuitive, but it is indeed a subset of any set because there are no elements in the empty set that could possibly not be in any other set. This fact reinforces the idea that even though the empty set contains nothing, it plays an integral role in the structure of set theory.