Analyzing Relation Properties: Reflexivity, Symmetry, Transitivity

Let's analyze the given relation '*' defined on positive numbers x and y. The relation is given by x * y if and only if \(x \le y^2\). We need to check if this relation possesses the properties of reflexivity, symmetry, and transitivity for all positive numbers x, y, and z.

Checking for Reflexivity of the Relation

A relation is reflexive if every element is related to itself. For the relation '*' on positive numbers, it is reflexive if for every positive number x, x * x holds true. This means we must check if \(x \le x^2\) for all positive x.

Let's test this condition with some positive values for x:

• If \(x = 1\), the condition is \(1 \le 1^2\), which is \(1 \le 1\). This is true.
• If \(x = 2\), the condition is \(2 \le 2^2\), which is \(2 \le 4\). This is true.
• If \(x = 0.5\) (which is a positive number), the condition is \(0.5 \le (0.5)^2\), which is \(0.5 \le 0.25\). This is false.

Since there exists a positive number (like 0.5) for which the condition \(x \le x^2\) is false, the relation '*' is not reflexive for all positive numbers.

Checking for Symmetry of the Relation

A relation is symmetric if whenever x is related to y, y is also related to x. For the relation '*', it is symmetric if for any positive numbers x and y, whenever \(x * y\) is true, \(y * x\) is also true. This means if \(x \le y^2\) is true, then \(y \le x^2\) must also be true for all positive x and y.

Let's test this condition with some positive values for x and y:

• Let \(x = 1\) and \(y = 2\). Is \(x * y\)? Is \(1 \le 2^2\)? Is \(1 \le 4\)? Yes, true. Now, is \(y * x\)? Is \(2 \le 1^2\)? Is \(2 \le 1\)? No, false.

Since we found a case where \(x * y\) is true (\(1 * 2\)) but \(y * x\) is false (\(2 * 1\)), the relation '*' is not symmetric for all positive numbers.

Checking for Transitivity of the Relation

A relation is transitive if whenever x is related to y and y is related to z, then x is also related to z. For the relation '*', it is transitive if for any positive numbers x, y, and z, whenever \(x * y\) is true and \(y * z\) is true, \(x * z\) must also be true. This means if \(x \le y^2\) and \(y \le z^2\), then \(x \le z^2\) must hold true for all positive x, y, and z.

Let's test this condition with some positive values for x, y, and z:

• Let \(x = 3\), \(y = 2\), and \(z = 1.5\). All are positive numbers.
• Is \(x * y\)? Is \(3 \le 2^2\)? Is \(3 \le 4\)? Yes, true. So \(x \le y^2\) holds.
• Is \(y * z\)? Is \(2 \le (1.5)^2\)? Is \(2 \le 2.25\)? Yes, true. So \(y \le z^2\) holds.
• Now, is \(x * z\)? Is \(3 \le (1.5)^2\)? Is \(3 \le 2.25\)? No, false. The condition \(x \le z^2\) does not hold.

Since we found a case where \(x * y\) and \(y * z\) are true, but \(x * z\) is false, the relation '*' is not transitive for all positive numbers.

Summary of Relation Properties

Based on our analysis:

• The relation '*' is not reflexive.
• The relation '*' is not symmetric.
• The relation '*' is not transitive.

Therefore, the relation does not possess any of the properties mentioned (reflexive, symmetric, transitive) for all positive numbers.

Conclusion based on options

Let's evaluate the given options:

• Option 1: is reflexive but not transitive and symmetric - Incorrect, as the relation is not reflexive.
• Option 2: is transitive but not reflexive and symmetric - Incorrect, as the relation is not transitive.
• Option 3: is symmetric and reflexive but not transitive - Incorrect, as the relation is neither symmetric nor reflexive.
• Option 4: None - This aligns with our finding that the relation is neither reflexive, symmetric, nor transitive for all positive numbers.

Revision Table: Key Relation Properties

Additional Information on Relation Types

Understanding relation properties like reflexivity, symmetry, and transitivity is fundamental in set theory and discrete mathematics. Relations that satisfy certain combinations of these properties are given special names:

• Equivalence Relation: A relation that is reflexive, symmetric, and transitive. These relations partition a set into disjoint equivalence classes. Example: Equality (=) on any set.
• Partial Order Relation: A relation that is reflexive, antisymmetric (if (a,b) and (b,a) are in the relation, then a=b), and transitive. Example: Less than or equal to (\(\le\)) on real numbers.

The relation \(x \le y^2\) on positive numbers demonstrates that a relation may not satisfy any of these standard properties, requiring careful verification based on the specific definition and the set it operates on.