Problem 28 - Discrete Mathematics with Applications

Question

Mark each as true or false, where \(A, B,\) and \(C\) are arbitrary sets and \(U\) the universal set. $$\varnothing-A=-A$$

Step-by-Step Solution

Answer

The statement $$\varnothing-A=-A$$ is \(\textbf{false}\). This is because the expressions \(\varnothing - A = \varnothing\) and \(-A = U - A\) are not identical. The empty set \(\varnothing\) has no elements, whereas the complement \(-A\) contains all elements in the universal set \(U\) that are not in \(A\).

Step 1: Find the set difference \(\varnothing - A\)

To find the set difference \(\varnothing - A\), we have to look for all elements that are in the empty set but not in A. However, by definition, there are no elements in the empty set. Thus, no elements can be in \(\varnothing - A\), so the set difference is the empty set itself. We can write this as: \[ \varnothing - A = \varnothing \]

Step 2: Find the complement set \(-A\)

The complement of set A, denoted as -A, consists of all elements in the universal set U that are not in A. We can write this as: \[ -A = U - A \]

Step 3: Compare the expressions

Now, let's compare the two expressions \(\varnothing - A\) and \(-A\). From Step 1 and Step 2, we derived the expressions: \[ \varnothing - A = \varnothing \] and \[ -A = U - A \] As we can see, the expressions are not identical: \(\varnothing\) is the empty set, which doesn't contain any elements, while \(-A\) contains all elements in the universal set U that are not in A. Thus, the given statement is false.

Key Concepts

Set Difference

In set theory, the concept of set difference is quite important. Set difference refers to the elements that are in one set but not in another. To put it simply, if you have two sets, say Set A and Set B, the difference between Set A and Set B (denoted as \( A - B \)) would be all elements that are only in Set A and not in Set B. This operation helps us see what is unique to one set when compared to another.

**Example:**
- Let \( A = \{ 1, 2, 3 \} \) and \( B = \{ 2, 3, 4 \} \). The set difference \( A - B = \{ 1 \} \), because 1 is the only element in A and not included in B.

Understanding set difference is vital because it lays the groundwork for more advanced concepts like complements and the empty set.

Empty Set

The empty set, denoted as \(\emptyset\), is a fundamental concept in set theory. It is the set that contains no elements at all. It might seem simple, but it is an essential idea that proves useful in various mathematical theories and operations.

Key attributes of the empty set are:

• Uniqueness: There is only one empty set, and no matter where you encounter it, it remains the same.


• Set operations: When we talk about operations involving the empty set, like \(\emptyset - A\), the result is always the empty set, since there are no elements in the empty set to "subtract" from any other set.

The empty set acts as a neutral element in many scenarios and helps reinforce the boundaries of a set by providing a baseline of no elements.

Universal Set

A universal set, often denoted by \( U \), encompasses all possible elements under consideration for a particular discussion or problem. The definition of a universal set can vary depending on what sets are being compared, but it always covers the complete range of elements relevant to the context.

Characteristics of a universal set include:

• Inclusiveness: It contains every single element in the discussion or the problem at hand.


• Comprehensive: Relative to specific sets you are working with, the universal set helps define operations such as unions, intersections, and complements.

In problems like finding the complement of a set or exploring set differences that involve all potential elements, the universal set is crucial because it serves as the "background" set that all other sets are measured against.

Complement of a Set

The complement of a set, often denoted as \(-A\) or \(U - A\), refers to all elements that are not in the set A but are within the universal set \(U\). This operation highlights what is "outside" the set when compared to everything available in the universal set.

To find the complement of a set:

• Identify all elements in the universal set \(U\).


• Exclude the elements found in set \(A\).

Mathematically, this operation is expressed as \(-A = U - A\), capturing everything not part of set A but still part of the broader universal set. This is essential for understanding how sets relate to each other within the larger context of all possible elements.