For Each Of These Relations On The Set {1, 2, 3, 4}, Decide Whether It Is Reflexive, Whether It Is Symmetric,

For Each Of These Relations On The Set {1, 2, 3, 4}, Decide Whether It Is Reflexive, Whether It Is Symmetric, understanding the properties of relations is fundamental in set theory and discrete mathematics. These properties help classify and analyze how elements within a set relate to each other, providing insights into the structure and behavior of the relations. In this article, we will explore various relations defined on the set {1, 2, 3, 4} and determine whether each relation is reflexive, symmetric, or possesses other notable properties. By doing so, we aim to develop a clear understanding of how to analyze relations systematically and to understand the significance of these properties in mathematical contexts.

Understanding Key Properties of Relations

Before delving into specific relations, it is essential to review the fundamental properties that characterize relations, especially reflexivity and symmetry.

Reflexive Relations

A relation R on a set S is called reflexive if every element is related to itself. Formally:

• For all a in S, (a, a) ∈ R.

This means that the diagonal elements of the relation matrix (if represented as such) are always present.

Symmetric Relations

A relation R on a set S is symmetric if, whenever an element a is related to an element b, then b is also related to a:

• For all a, b in S, if (a, b) ∈ R, then (b, a) ∈ R.

This property indicates a mutual relationship between related pairs.

Analyzing Specific Relations on the Set {1, 2, 3, 4}

Let's consider several relations defined on the set {1, 2, 3, 4} and analyze their properties.

Relation R1: The "Less Than" Relation

• Definition: R1 = { (a, b) | a < b } on {1, 2, 3, 4}.

Evaluation:

• Reflexive?
• For a relation to be reflexive, (a, a) must be in R for all a.
• Since a < a is never true, (a, a) is not in R1 for any a.
• Conclusion: R1 is not reflexive.
• Symmetric?
• Suppose (a, b) ∈ R1, which means a < b.
• Is (b, a) in R1? Since b < a would be needed, which is false if a < b.
• Therefore, the relation does not contain both (a, b) and (b, a) unless a = b, which it does not.
• Conclusion: R1 is not symmetric.

Summary for R1:

• Reflexive? No
• Symmetric? No

Relation R2: The "Equal To" Relation

• Definition: R2 = { (a, b) | a = b }.

Evaluation:

• Reflexive?
• For all a, (a, a) ∈ R2.
• Yes, since every element equals itself.
• Conclusion: R2 is reflexive.
• Symmetric?
• If (a, b) ∈ R2, then a = b.
• It follows that (b, a) = (a, b), so (b, a) ∈ R2.
• Conclusion: R2 is symmetric.

Summary for R2:

• Reflexive? Yes
• Symmetric? Yes

Relation R3: The "Divides" Relation

• Definition: R3 = { (a, b) | a divides b }.

Evaluation:

• Reflexive?
• Every number divides itself, so (a, a) ∈ R3 for all a.
• Conclusion: R3 is reflexive.
• Symmetric?
• Suppose (a, b) ∈ R3, meaning a divides b.
• Is (b, a) in R3? Only if b divides a.
• For example, (2, 4) ∈ R3 (since 2 divides 4), but (4, 2) ∉ R3 (4 does not divide 2).
• Conclusion: R3 is not symmetric.

Summary for R3:

• Reflexive? Yes
• Symmetric? No

Relation R4: The "Is a Friend of" Relation (Assuming Symmetry)

• Definition: R4 = { (a, b) | a is a friend of b }.

Suppose this relation is symmetric by nature (friendship is generally mutual).

Evaluation:

• Reflexive?


• Does everyone consider themselves a friend? Usually, in social contexts, yes.


• Assuming yes: (a, a) ∈ R4 for all a.


• Conclusion: R4 is reflexive.


• Symmetric?


• Friendship is mutual, so if (a, b) ∈ R4, then (b, a) ∈ R4.


• Conclusion: R4 is symmetric.

Summary for R4:

• Reflexive? Yes (assuming self-friendship)


• Symmetric? Yes

Additional Relations and Their Properties

Beyond the basic relations above, other relations can be constructed or analyzed to understand their properties.

Relation R5: The "Greater Than" Relation

• Definition: R5 = { (a, b) | a > b }.

Evaluation:

• Reflexive?
• Since a > a is false, (a, a) ∉ R5, so not reflexive.
• Symmetric?
• If a > b, then b > a is false, so (b, a) ∉ R5.
• Not symmetric.
• Summary: Not reflexive, not symmetric.

Relation R6: The "Square" Relation

• Definition: R6 = { (a, b) | a² = b }.

Evaluation:

• Reflexive?
• For a to be related to itself, a² = a.
• Let's check for each element:
• 1: 1² = 1 → yes
• 2: 4 ≠ 2 → no
• 3: 9 ≠ 3 → no
• 4: 16 ≠ 4 → no
• Only (1, 1) belongs.
• Not all elements are related to themselves.
• Conclusion: Not reflexive.
• Symmetric?
• For symmetry, if (a, b) ∈ R6, then (b, a) ∈ R6.
• (a, b) ∈ R6 implies a² = b.
• For (b, a) to be in R6, b² = a.
• Combining these:
• a² = b and b² = a → a² = b and (a²)² = a → a⁴ = a.
• For a ∈ {1, 2, 3, 4}, check which satisfy a⁴ = a:
• 1: 1⁴ = 1 → yes
• 2: 2⁴ = 16 ≠ 2 → no
• 3: 81 ≠ 3 → no
• 4: 256 ≠ 4 → no
• Only a=1 satisfies this.
• So (1, 1) is in R6, but not necessarily the others.
• Thus, the relation is not symmetric.

Summary Table of Relations and Properties

| Relation | Is Reflexive? | Is Symmetric? | Comments |
|------------|----------------|----------------|----------|
| R1: Less Than | No | No | Strict inequality |
| R2: Equal To | Yes | Yes | Identity relation |
| R3: Divides | Yes | No | Divisibility relation |
| R4: Friendship | Yes | Yes | Assumed mutuality |
| R5: Greater Than | No | No | Opposite of R1 |
| R6: Square Relation | No | No | Based on squares |

Conclusion

Analyzing relations on a finite set like {1, 2, 3, 4} provides valuable insights into their structure and properties. Recognizing whether a relation is reflexive or symmetric helps classify the relation and understand its behavior. For example, the equality relation is both reflexive and symmetric, typical of equivalence relations. In contrast, the "less than" and "greater than" relations are neither reflexive nor symmetric, but they are transitive, which could be analyzed further.

Understanding these properties is crucial in fields such as computer science, mathematics, and logic, where relations underpin structures like graphs, equivalence classes, and ordering systems. By systematically evaluating each relation's properties, one can