Problem 2 - A Gentle Introduction to the Art of Mathematics

Question

Consider the relation A defined by \(\mathrm{A}=\\{(x, y) \mid x\) has the same astrological sign as \(y\\}\). Is A symmetric or anti-symmetric? Explain.

Step-by-Step Solution

Answer

The relation A is symmetric but not anti-symmetric.

Step 1: Identify the Relation

The relation A consists of pairs \((x, y)\) where x and y share the same astrological sign.

Step 2: Definition of Symmetric Relation

A relation is symmetric if \((x, y)\) being in the relation implies that \((y, x)\) is also in the relation.

Step 3: Check for Symmetry

For \((x, y)\) to be in the relation A, x and y must have the same astrological sign. This implies that y and x also share the same astrological sign, hence \((y, x)\) is also in A. Therefore, A is symmetric.

Step 4: Definition of Anti-Symmetric Relation

A relation is anti-symmetric if \((x, y)\) and \((y, x)\) being in the relation implies that x=y.

Step 5: Check for Anti-Symmetry

Consider any pair \((x, y)\) and \((y, x)\) in A. This only tells us that x and y have the same astrological sign, not necessarily that x=y. Therefore, the relation A is not anti-symmetric.

Key Concepts

Relation

In mathematics, a relation defines a connection between elements of two sets. If you have two sets, say Set X and Set Y, a relation between them consists of ordered pairs \( (x, y) \) where x is from Set X and y is from Set Y. For example, we might say x is related to y if they share something in common, like an astrological sign. This connection can be described by various properties which help understand the characteristics of the relation.

Symmetric Relation

A symmetric relation is a type of relation where if an element x is related to an element y, then y is also related to x. In mathematical terms, if \( (x, y) \) is in the relation, then \( (y, x) \) must also be in the relation.
To put it simply:
If John has the same astrological sign as Mary, then Mary must also have the same sign as John.
This property makes the relation symmetric.
In the context of our exercise, since having the same astrological sign is mutual, the relation is indeed symmetric.

Anti-Symmetric Relation

An anti-symmetric relation, on the other hand, is quite different. This type of relation says if \( (x, y) \) and \( (y, x) \) are both in the relation, then x must be equal to y. In simpler terms:
If John is related to Mary and Mary is related to John, then John must be Mary (which obviously can't happen because they are different people).
Consider the relations x \( eq \) y and y \( eq \) x from our exercise. This does not necessarily tell us that x = y. Thus, the relation does not meet the criteria for anti-symmetric properties.

Mathematical Logic

To understand relations better, we often use mathematical logic, which provides a framework to reason through such problems systematically. Using logical steps, we can determine various properties of relations like symmetry and anti-symmetry.
Here’s how logic helps:

• We list out known properties and definitions.


• Check if the relation fits these definitions.


• Apply logical reasoning to verify the property.

In the given exercise, by breaking down the problem into smaller steps and applying definitions of symmetric and anti-symmetric relations, we were able to conclude that the relation based on common astrological signs is symmetric but not anti-symmetric.