Problem 33 - Discrete Mathematics with Applications

Question

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$(A \cup B)^{\prime}=A^{\prime} \cup B^{\prime}$$

Step-by-Step Solution

Answer

Based on the provided step-by-step solution, the short answer is: The given statement \(\left(A^{\prime}\right)^{\prime}=A\) is *True*. This is because when we find the complement of the complement of set A, it simplifies to the original set A itself.

Step 1: Understand the complement of a set

The complement of a set A (denoted as \(A'\) or \(A^c\)) contains all the elements that are not in set A but are in the universal set U. In other words, \(A' = U - A\). The given statement states that \(\left(A^{\prime}\right)^{\prime}=A\). We need to find out if this is true.

Step 2: Analyzing the properties of complements of sets

Let's analyze the left-hand side of the given equation \(\left(A^{\prime}\right)^{\prime}\). Here, we are taking the complement of the complement of set A. As a first step, we know that \(A^{\prime} = U - A\). Now, let's find the complement of this set: \(\left(U - A\right)^{\prime} = U - (U - A)\)

Step 3: Simplify the expression

Now let's simplify the given expression to find the result. \(\left(U - A\right)^{\prime} = U - (U - A) = U - U + A\) Since the difference between the universal set and itself is an empty set, we are left with: \(\left(U - A\right)^{\prime} = \emptyset + A = A\)

Step 4: Conclusion

Comparing the original given statement \(\left(A^{\prime}\right)^{\prime}=A\) with our derived expression in Step 3, we found that they are the same. Therefore, the given statement is *True*.

Key Concepts

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. It is one of the fundamental languages of mathematics and provides a foundation for various other fields, including discrete mathematics. A set is a collection of distinct objects, considered as an object in its own right. For instance, a set can consist of numbers, letters, or even other sets.

In the context of the provided exercise, we encounter the concept of set complement. The complement of a set, denoted as \(A'\) or \(A^c\), includes every element in the universal set \(U\) that is not in \(A\). This operation is fundamental in understanding how sets interact with each other within the confines of a universal set, which contains all possible elements of discussion in a particular context.

Universal Set

The universal set is a set that contains all the objects we are interested in for a particular discussion or problem. This set is often denoted by \(U\) and it can be finite or infinite depending on context. The universal set is crucial in the operation of complements because it serves as the reference point for what elements are included in the complement.

In our example, the complement of set \(A\) was defined relative to the universal set with the expression \(A' = U - A\). This essentially means that when we are discussing complements, we are always referring to what's left in the universal set after removing the set in question. The complement of the complement operation brings us back to our original set, proving the given statement true.

De Morgan's Laws

De Morgan's laws are fundamental rules in set theory and logic. They connect the operations of conjunction (and) and disjunction (or) with the not operation in terms of set complement. There are two De Morgan's laws that pertain to sets:

• \(\overline{A \cup B} = \overline{A} \cap \overline{B}\)
• \(\overline{A \cap B} = \overline{A} \cup \overline{B}\)

These laws show how the complement of a union of sets is the intersection of their complements, and vice versa.

Although De Morgan's laws are not directly used in the given exercise, understanding these rules helps in grasping the underlying principles of how set operations interrelate and provides a broader perspective on the behavior of complements in set theory.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This includes topics such as set theory, logic, graph theory, and combinatorics. It deals with objects that can assume a distinct and separate form.

Set theory is a key part of discrete mathematics since the objects dealt with in discrete structures can often be grouped into sets. The exercises that we work on, such as calculating the complement and understanding the relationship between different sets, are part of the discipline's fundamental concepts. Discrete mathematics has wide applications in computer science, cryptography, and algorithm design, making it essential for students in those fields to master the basics like set operations and properties.