Problem 1 - A Gentle Introduction to the Art of Mathematics

Question

Consider the relation \(\mathrm{S}\) defined by \(\mathrm{S}=\\{(x, y) \mid x\) is smarter than \(y\\}\). Is S symmetric or anti-symmetric? Explain.

Step-by-Step Solution

Answer

The relation \(\text{S}\) is anti-symmetric because if \(x\) is smarter than \(y\), \(y\) cannot be smarter than \(x\).

Step 1: Understand the Definition of the Relation S

The relation \(\text{S} = \{(x, y) \mid x \text{ is smarter than } y\}\) implies that \(x\) is smarter than \(y\). This indicates a comparison between two individuals based on their intelligence.

Step 2: Determine Symmetry

A relation \(R\) is symmetric if \( (x, y) \in R \rightarrow (y, x) \in R \). In this case, if \(x\) is smarter than \(y\), it does not imply that \(y\) is smarter than \(x\). Therefore, the relation \(\text{S}\) is not symmetric.

Step 3: Determine Anti-Symmetry

A relation \(R\) is anti-symmetric if \( (x, y) \in R \text{ and } (y, x) \in R \rightarrow x = y \). In this case, if \(x\) is smarter than \(y\) and \(y\) is smarter than \(x\), then \(x = y\). There's no logical situation where one is smarter than the other and vice versa simultaneously unless they are the same person. Thus, \(\text{S}\) is anti-symmetric.

Step 4: Conclusion

Since the relation \(\text{S}\) is not symmetric and is anti-symmetric, the relation fits the criteria of being anti-symmetric.\r

Key Concepts

Symmetric Relation

A symmetric relation is a type of relation in mathematics where if a pair \(x, y\) is in the relation, then the pair \(y, x\) must also be in the relation. For example, if \(x\) is related to \(y\), then \(y\) must be related to \(x\) for the relation to be symmetric.

Think of it like a two-way street. If you can drive from point A to point B, you must also be able to drive back from point B to point A for it to be called symmetric.

Considering the provided exercise, the relation S = \{(x, y) \mid x is smarter than y\} implies that if \(x\) is smarter than \(y\), the reverse, \(y\) being smarter than \(x\), doesn’t hold true. Therefore, the relation is not symmetric.

Anti-Symmetric Relation

An anti-symmetric relation, meanwhile, is a relation where, if \(x\) is related to \(y\) and \(y\) is also related to \(x\), then \(x\) must be equal to \(y\).

For instance, if in a mathematical relation we have both \(x, y\) and \(y, x\) as true, it logically concludes that \(x=y\).

In the context of our exercise question about the relation S = \{(x, y) \mid x is smarter than y\}, if \(x\) is stated to be smarter than \(y\) and simultaneously \(y\) is smarter than \(x\), the only possibility would be if \(x\) and \(y\) are indeed the same person, implying no contradiction. Hence, this relation fits the criteria of being anti-symmetric.

Mathematical Logic

Mathematical logic is the discipline that captures the principles of valid reasoning and formal mathematical proofs. It underpins the methods used to derive conclusions from axioms and established truths.

It involves:

• Understanding definitions
• Analyzing relationships
• Making logical inferences

Regarding our problem with the relation S and its properties, mathematical logic helps us rigorously decide if the relation is symmetric or anti-symmetric. By utilizing precise definitions and logical deductions, we concluded that the relation is not symmetric but is anti-symmetric. Thus, mathematical logic provides the framework to reach these conclusions in a structured manner.