Problem 34 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The statement \((\boldsymbol{A}^{\prime})^{\prime}=\boldsymbol{A}\) is true since, by definition of the complement, an element x is in the set A if and only if it is also in \((A')'\). This verifies the equality \((A')' = A\).

Step 1: Define the Complement of a Set

The complement of a set A, denoted as \(A'\) (or sometimes as \(A^C\)), is defined as the set of all elements that exist in the universal set (U) but not in set A. Formally, \(A' = \{x \in U : x \notin A\}\). Now we have to prove if \((A')' = A\).

Step 2: Define the Complement of the Complement of a Set

We need to find the complement of the complement of set A, which is denoted as \((A')'\). Formally, \((A')' = \{x \in U : x \notin A'\}\).

Step 3: Prove the Equality

Our goal is to prove that \((A')' = A\). If an element x belongs to \((A')'\), that means x is not in \(A'\), which in turn means that x must be in set A (because if a member of the universal set is not in the complement of A, then it must be in A). Hence, x is in A. Now suppose that an element x is in set A. Since the element is in A, it cannot be in its complement, i.e., x cannot belong to \(A'\). This means x is in the complement of \(A'\), which is \((A')'\). Therefore, in both cases, element x is in the set A if and only if it is also in \((A')'\). This verifies the equality \((A')' = A\). As we demonstrated that taking the complement of a complement of a set indeed equals the original set, the given statement is true.

Key Concepts

complement of a set

In set theory, the complement of a set is an important concept. Think of it as the "opposite" of a set. The complement of a set \(A\), denoted as \(A'\), is made up of all elements that belong to the universal set \(U\) but are not found in set \(A\).
This idea helps us differentiate between what is and isn't part of a specific collection within a larger pool. It's like identifying all the puzzle pieces that don't fit a particular space in the universal puzzle.

To express this formally, we write:
\[A' = \{x \in U : x otin A\}\].
This notation means "the set of all elements \(x\) such that \(x\) is in the universal set \(U\) and \(x\) is not in the set \(A\)."

• Key understanding: \(A'\) gives us a clear and distinct view of everything outside the set \(A\).
• Useful for analyzing relationships between sets.

Understanding complements is foundational in set theory, often leading to more complex operations and theorems.

universal set

The universal set, denoted as \(U\), is the entire collection of possible elements under consideration within a given context. Every set in a specific discussion or problem is typically a subset of this universal set.
It serves as the backdrop against which all sets are compared and evaluated.

The universal set is like considering all students in a school if you're talking about different classes or groups within that school. Here are a few essential points to grasp:

• Every relevant element is included in \(U\), so the complement of any set will also be drawn from \(U\).
• The choice of \(U\) can change based on context or the nature of the problem. For a math class, it could be all numbers; for biology, it might be all species.

The flexibility and comprehensive nature of the universal set make it a crucial tool in performing set operations and understanding set relationships.

set operations

Set operations are ways to manipulate and combine sets to explore relationships and solve problems. Common operations include union, intersection, and complement, each with its own rules and properties.

Let's break down a few fundamental set operations:

• Intersection: The intersection of two sets \(A\) and \(B\) is the set of elements that are common to both, denoted as \(A \cap B\). For example, if \(A = \{1, 2, 3\}\) and \(B = \{2, 3, 4\}\), then \(A \cap B = \{2, 3\}\).
• Union: The union of two sets \(A\) and \(B\) consists of all elements that are in \(A\), \(B\), or in both, written as \(A \cup B\).
• Complement: As discussed earlier, involves elements in the universal set that are not in the particular set.

These operations help us in numerous ways:

• They allow for the representation of complex relationships and scenarios.
• They form the basis for further mathematical theories and proofs across different fields of study.

Mastering set operations opens doors to a deeper understanding of mathematical logic and structure.