Analyzing Relation Properties based on Logarithms

The question asks us to determine the properties (reflexive, symmetric, transitive) of a relation R defined on the set S × S, where S = {1, 2, 3, ...}. The relation is given by xRy if \(\log_a x > \log_a y\) with \(a = \frac{1}{2}\).

Understanding the Relation Condition

The given condition for xRy is \(\log_{1/2} x > \log_{1/2} y\). To understand this inequality, we need to consider the base of the logarithm. The base is \(a = \frac{1}{2}\), which is between 0 and 1 (\(0 < \frac{1}{2} < 1\)).

For a logarithm with a base \(b\) such that \(0 < b < 1\), the logarithmic function \(f(z) = \log_b z\) is a decreasing function. This means that if \(\log_b u > \log_b v\), then it must be true that \(u < v\).

Applying this to our relation, xRy holds if and only if \(x < y\), given that \(x\) and \(y\) are elements of S = {1, 2, 3, ...}, so they are positive integers.

Thus, the relation R is simply the "less than" relation defined on the set of natural numbers S.

Checking Relation Properties

Now let's check if this relation R (where xRy means x < y) is reflexive, symmetric, or transitive on S.

Reflexive Property Analysis

A relation R on a set S is reflexive if for every element \(x \in S\), \(xRx\) holds.

For our relation R, \(xRx\) would mean \(x < x\). Is this true for any \(x \in S\)?

• Take an element from S, for example, 5. Does 5R5 hold? This means \(5 < 5\). This is false.
• In general, for any number \(x\), \(x\) is never less than itself.

Therefore, the relation R is not reflexive.

Symmetric Property Analysis

A relation R on a set S is symmetric if for every \(x, y \in S\), whenever \(xRy\) holds, \(yRx\) also holds.

For our relation R, if \(xRy\) holds, it means \(x < y\). If R were symmetric, then \(yRx\) would also have to hold, meaning \(y < x\).

• Take two distinct elements from S, for example, 3 and 7.
• Does 3R7 hold? Yes, because \(3 < 7\).
• If R were symmetric, 7R3 would have to hold. Does 7R3 hold? This means \(7 < 3\), which is false.
• Since we found a pair (3, 7) where 3R7 holds but 7R3 does not hold, the relation is not symmetric.

Therefore, the relation R is not symmetric.

Transitive Property Analysis

A relation R on a set S is transitive if for every \(x, y, z \in S\), whenever \(xRy\) holds and \(yRz\) holds, \(xRz\) also holds.

For our relation R, if \(xRy\) holds, it means \(x < y\). If \(yRz\) holds, it means \(y < z\). If R is transitive, then \(xRz\) must hold, meaning \(x < z\).

• Let's consider elements \(x, y, z \in S\).
• Suppose \(xRy\) is true, so \(x < y\).
• Suppose \(yRz\) is true, so \(y < z\).
• We need to check if \(xRz\) (i.e., \(x < z\)) is always true when \(x < y\) and \(y < z\).
• This is a fundamental property of the "less than" relation on numbers. If \(x\) is less than \(y\), and \(y\) is less than \(z\), then \(x\) must be less than \(z\).
• For example, if 2 < 5 and 5 < 9, then it is true that 2 < 9.

This property holds true for all elements in S. Therefore, the relation R is transitive.

Conclusion on Relation Properties

Based on our analysis:

• The relation R (xRy iff x < y) is not reflexive.
• The relation R (xRy iff x < y) is not symmetric.
• The relation R (xRy iff x < y) is transitive.

So, the relation is transitive only.

Evaluating the Options

Let's look at the given options:

1. reflexive only (Incorrect, as it is not reflexive)
2. symmetric only (Incorrect, as it is not symmetric)
3. transitive only (Correct, as it is transitive but not reflexive or symmetric)
4. both symmetric and transitive (Incorrect, as it is not symmetric)

Revision Table: Relation Properties

Additional Information: Logarithm Properties and Relations

Understanding how the base of a logarithm affects inequalities is crucial here. For \(\log_b u > \log_b v\):

• If \(b > 1\), this implies \(u > v\). Examples: \(\log_{10} 100 > \log_{10} 10\) implies \(100 > 10\).
• If \(0 < b < 1\), this implies \(u < v\). Examples: \(\log_{1/2} 8 > \log_{1/2} 4\) is \(-3 > -2\), which is false. So \(\log_{1/2} 4 > \log_{1/2} 8\) is \(-2 > -3\), which is true, and it implies \(4 < 8\).

Relations are fundamental concepts in mathematics used to describe how elements within a set or between sets are connected. Properties like reflexivity, symmetry, and transitivity help classify relations and are important in various areas like set theory, graph theory, and database theory. An equivalence relation, for instance, must be reflexive, symmetric, and transitive. A partial order relation must be reflexive, antisymmetric, and transitive.