Problem 31 - Discrete Mathematics with Applications

Question

Find the power set of each set. $$\\{\mathrm{a}\\}$$

Step-by-Step Solution

Answer

The power set of the given set \(\{\mathrm{a}\}\) is \(P(\{\mathrm{a}\}) = \{ \varnothing, \{\mathrm{a}\} \}\).

Step 1: List all possible subsets of the given set

The given set has only one element: {a}. The possible subsets of this set are - the empty set and the set containing "a" itself. So we have the following subsets: 1. The empty set: 𝜑 or { } 2. The set containing "a": {a}

Step 2: Combine all subsets into the power set

Now that we have listed all possible subsets, we can combine them to form the power set. The power set is the set of all the subsets we listed above. Therefore, the power set of the given set {a} is: \[ P(\{\mathrm{a}\}) = \{ \varnothing, \{\mathrm{a}\} \} \] So the power set of the given set is { 𝜑, {a} }.

Key Concepts

Discrete Mathematics and Power Sets

Discrete mathematics is a branch of mathematics dealing with discrete elements that use algebra and arithmetic. A core component within discrete mathematics is the concept of a power set. A power set is a term that describes all possible subsets of a given set, where each subset is a combination of elements from the original set.

For instance, the exercise provided showcases the simplest form of a power set where the original set contains only one element, \( \{a\} \). The power set, in this case, is composed of the empty set and the set containing only the element 'a'. Understanding how to construct a power set is fundamental in discrete mathematics because it forms the basis for more complex topics such as combinatorics and logic.

Subsets

A subset is a set where all elements of the subset are also contained within another set, known as the superset. In other words, if we have two sets, A and B, then B is a subset of A (\( B \subseteq A \) if every element of B is also an element of A.

In the context of our exercise, there are two subsets of the set \( \{a\} \): the empty set \( \varnothing \), which is a subset of every set, and the set itself \( \{a\} \). It's important to recognize that subsets include not only combinations of elements but also the original set and even the absence of elements (the empty set).

Set Theory

Set theory is the mathematical study of sets, which are collections of objects. It serves as the foundation for various areas of mathematics. Sets are often denoted with curly braces, for example \( \{a, b, c\} \), and involve operations such as union, intersection, and complement.

The exercise we are looking at touches the fundamental concept in set theory: the power set. Set theory introduces the idea that for any given set, there is a collection of all possible subsets—including the set itself and the empty set—that can be generated, which is what we refer to as the power set. The power set is a larger set containing all these combinations and is a crucial concept for students to understand.