Problem 19 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The given statement is false, as the set \(\{\emptyset\}\) contains one element (the empty set itself), while the empty set \(\varnothing\) has no elements. Therefore, \(\{\emptyset\} \neq \varnothing\).

Step 1: Comparing the sets

Compare the set containing the empty set: \(\{\emptyset\}\) with the empty set: \(\varnothing\). The set \(\{\emptyset\}\) has one element, which is the empty set itself. On the other hand, the empty set, \(\varnothing\), has no elements.

Step 2: Determine if the statement is true or false

Since the set \(\{\emptyset\}\) has one element while the empty set \(\varnothing\) has no elements, we can conclude that the given statement is false. So, the answer is: \(\{\emptyset\} \neq \varnothing\) (False).

Key Concepts

Empty Set

In set theory, the concept of an empty set is fundamental. An empty set, denoted by \(\varnothing\) or \(\{\}\), is a unique set that contains no elements at all. It's like an empty box that holds nothing inside. This is important because it represents the concept of "none" in set terms, akin to zero in arithmetic. There are no items or members in the empty set. It is also the only set with zero cardinality, which refers to the number of elements in a set.
Unlike other sets that can contain numbers or objects, the empty set stands apart due to its lack of contents. This makes it a subset of every other set by virtue of having no elements to conflict with other sets. In many mathematical contexts, the empty set plays a crucial role by serving as a base case or starting point.

Elements of a Set

To understand the structure of a set, we must understand its elements. The elements of a set are the distinct objects or members that it comprises. These elements can be numbers, letters, or other objects, and they are usually enclosed within curly braces, such as \(\{1, 2, 3\}\).
It's important to note that in a set:

• The order of elements does not matter, so \(\{a, b, c\} = \{c, b, a\}\).
• Each element is unique, meaning no duplicates are allowed.
• Elements could even be other sets themselves.

When examining the set \(\{\emptyset\}\) from the exercise, it consists of one element: the empty set itself. This demonstrates how sets can contain other sets as elements. Understanding the elements of a set is crucial for comparing two sets to determine their equality.

Equality of Sets

In set theory, two sets are considered equal if they contain the exact same elements. Equality of sets is denoted by the symbol \(=\). To determine if two sets are equal, we must ensure:

• Every element of the first set is in the second set.
• Every element of the second set is in the first set.

In mathematical terms, this can be interpreted as two-way membership. For example, if \(A = \{1, 2\}\) and \(B = \{2, 1\}\), then \(A = B\) because they contain the same elements, irrespective of order.
When comparing \(\{\emptyset\}\) and \(\varnothing\) as in the original exercise, the set \(\{\emptyset\}\) and the empty set \(\varnothing\) are not equal. This is because \(\{\emptyset\}\) contains exactly one element: the empty set, while \(\varnothing\) contains no elements at all. Thus, these two sets do not meet the criteria for equality. Therefore, understanding the equality of sets requires a careful examination of their elements to ensure they match exactly.