Problem 27 - Discrete Mathematics with Applications

Question

Mark each as true or false. Every nonempty set has at least two subsets.

Step-by-Step Solution

Answer

The statement "Every nonempty set has at least two subsets" is true. A nonempty set always has at least the empty set (Ø) and the set itself as subsets. In cases where there are more elements in the set, there are even more possible subsets.

Step 1: Breaking down the statement

First, let's recall the definition of a subset. A subset is a set in which all its elements are also elements of another set. In other words, if A is a subset of B, then every element of A is also an element of B. Now let's focus on the statement. We need to determine if every nonempty set has at least two subsets. We can do this by looking at the possible subsets for different types of sets.

Step 2: Subset of a single-element set

Consider a nonempty set that has only one element, say A = {a}. In this case, there are two possible subsets: 1. The empty set: {} or Ø 2. The set itself: {a} Given that the set A has at least two subsets in this case, this part of the statement seems true.

Step 3: Subset of a multiple-element set

Now, let's consider a nonempty set that has more than one element, say B = {a, b}. In this case, there are four possible subsets: 1. The empty set: {} or Ø 2. A subset with just element 'a': {a} 3. A subset with just element 'b': {b} 4. The set itself: {a, b} Again, this part of the statement is also true as a multiple-element set has at least two subsets.

Step 4: Conclusion

The statement "Every nonempty set has at least two subsets" is true. As we demonstrated, a nonempty set always has at least the empty set and the set itself as subsets. In cases where there are more elements in the set, there are even more possible subsets.

Key Concepts

Subset Definition

The concept of a subset is foundational in the study of sets in mathematics. A subset of a set is defined as a set in which every element is also contained within another set, known as the superset. For instance, if we have a set called B, and there is another set A such that every element in A is also in B, we say A is a subset of B, denoted as \( A \subseteq B \).

It's important to know that by this definition, every set is a subset of itself because all the elements of a set are contained within the set. Furthermore, the empty set, denoted as \( \emptyset \) or \( \{\} \), is a subset of every set. This is because there are no elements in the empty set to contradict the definition of a subset. Therefore, it is correct to state that every nonempty set has the empty set and itself as subsets at the minimum.

Single-Element Set

When we talk about a single-element set, we mean a set that contains exactly one element. It is as simple as it sounds. For example, if we have set \( C = \{x\} \), where x is that one and only element, then C is a single-element set.

With regards to subsets, a single-element set always has two subsets - the set itself, which contains that one element, and the empty set. The empty set represents the idea of 'having nothing,' while the set \( C \) represents the concept of 'having that one element, x'. Thus, the statement 'Every nonempty set has at least two subsets' holds true for single-element sets. To ensure clarity, let's illustrate the subsets of \( C \):

• The empty set: \( \emptyset \)
• The set itself: \( C = \{x\} \)

Multiple-Element Set

A multiple-element set is a set that consists of more than one distinct element. Such sets offer a richer variety of subsets. Take, for instance, the set \( D = \{m, n\} \). As the set contains two elements, m and n, the subsets include combinations of these elements, as well as the empty set and the set D itself.

For a set with two elements, the possible subsets are:

• The empty set: \( \emptyset \)
• A subset with just the first element: \( \{m\} \)
• A subset with just the second element: \( \{n\} \)
• The set itself: \( D = \{m, n\} \)

Thus, the more elements a set has, the greater the number of subsets. In general, a set with n elements will have \( 2^n \) subsets, representing all possible combinations of the set’s elements. This further supports the assertion that every nonempty set has at least two subsets, with both multiple-element sets and single-element sets adhering to this rule.