Problem 22 - Discrete Mathematics with Applications

Question

Mark each as true or false. $$\\{x | x \neq x\\}=\varnothing$$

Step-by-Step Solution

Answer

The given set \(\\{x | x \neq x\\}\) contains all elements "x" for which the condition "x is not equal to x" holds true. However, this statement contradicts the reflexive property of equality, which states that every element is equal to itself (x = x). As there is no element that satisfies the condition "x ≠ x," the given set does not contain any elements, making it equal to the empty set (∅). Therefore, the mathematical statement is true.

Step 1: Understand the given set

The given set can be expressed as: \(\\{x | x \neq x\\}\). This means that the set contains all elements "x" for which the condition "x is not equal to x" holds true.

Step 2: Analyze the condition

The condition states that x is not equal to x (x ≠ x). This statement contradicts the reflexive property of equality, which states that every element is equal to itself (x = x). Therefore, there is no element that satisfies the condition "x ≠ x".

Step 3: Compare the given set with the empty set

Since there is no element that satisfies the condition "x ≠ x", the given set does not contain any elements. The empty set (∅) also does not contain any elements.

Step 4: Make a conclusion

As the given set does not contain any elements and the empty set (∅) also does not contain any elements, they can be considered equal. Therefore, the mathematical statement is true.

Key Concepts

Set Notation

Set notation is a precise way of describing a collection of objects. In mathematics, sets are defined by their elements. A set can be described by listing its elements inside curly braces, such as \( \{1, 2, 3\} \), or by stating a property that its members satisfy, like \( \{ x | x \text{ is even} \} \).

In the context of the exercise, the set \( \{x | x eq x\} \) is described using a property: it includes all elements \( x \) for which \( x eq x \). This is a kind of set-builder notation, where properties define the set’s elements.

Since no element can be unequal to itself, this set effectively includes no elements, which is why it is equal to the empty set, denoted as \( \varnothing \).

Reflexive Property

The reflexive property is a fundamental concept in mathematical logic referring to equality. It states that any element is equal to itself. For example, \( x = x \) is always true, regardless of what \( x \) represents.

In the problem at hand, you encounter the condition \( x eq x \). This contradicts the reflexive property because it suggests an element cannot be equal to itself, which is logically impossible in a standard sense.

Understanding the reflexive property helps us deduce that no element in the universe can satisfy \( x eq x \), leading to the conclusion that the described set is indeed empty.

Mathematical Logic

Mathematical logic is a foundation for reasoning mathematically. It involves the use of logical statements and conditions to deduce truths.

In this exercise, you are required to analyze a logical condition: \( x eq x \). From a logical perspective, this condition is inherently false for any real element because it violates the reflexive property of equality.

This logical analysis helps in understanding why \( \{x | x eq x\} \) results in no elements. Logical reasoning ensures that mathematical conclusions derived from such conditions are consistent and sound.

Equality in Mathematics

Equality is a basic concept in mathematics that indicates two expressions represent the same quantity or object. The notation \( = \) expresses this relationship.

Equating any number or element with itself is a simple and true expression of equality: \( x = x \). However, in scenarios like \( x eq x \), equality is challenged, making it impossible under usual circumstances since no real number can be unequal to itself.

Recognizing these principles of equality is crucial in solving and understanding mathematical problems, particularly those involving empty sets and conditions that appear to deny the truth of equality among elements.