Problem 2 - A Primer of Real Analysis

Question

Show that the relation \(R\) on \(\mathbb{Z}\) defined by \(m \sim_{R} n\) if \(m-n\) is even is an equivalence relation.

Step-by-Step Solution

Answer

The relation is an equivalence relation on \( \mathbb{Z} \).

Step 1: Identify the Requirements for an Equivalence Relation

Recall that an equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and transitivity. We'll prove that the relation \( R \), where \( m \sim n \) iff \( m - n \) is even, satisfies these properties on \( \mathbb{Z} \).

Step 2: Verify Reflexivity

For reflexivity, we need to show that for any integer \( m \in \mathbb{Z} \), \( m \sim m \). Calculate \( m - m = 0 \), and since 0 is even, this implies \( m \sim m \) is true. Hence, the relation is reflexive.

Step 3: Verify Symmetry

For symmetry, assume \( m \sim n \), meaning \( m - n \) is even. Thus, there exists some integer \( k \) such that \( m - n = 2k \). Consequently, \( n - m = -2k \), which is also even. Therefore, \( n \sim m \), and the relation is symmetric.

Step 4: Verify Transitivity

For transitivity, assume \( m \sim n \) and \( n \sim p \). Thus, \( m - n = 2k \) and \( n - p = 2l \) for some integers \( k \) and \( l \). Adding these, \( m - p = (m - n) + (n - p) = 2k + 2l = 2(k + l) \), which is even. This proves \( m \sim p \), showing the relation is transitive.

Step 5: Conclude the Relation is an Equivalence Relation

Since the relation \( R \) satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation on the set of integers \( \mathbb{Z} \).

Key Concepts

Understanding Reflexivity

To show that a relation is an equivalence relation, the first thing we must verify is reflexivity. Reflexivity implies that every element is related to itself.
For instance, consider any integer \( m \) in the set \( \mathbb{Z} \). To satisfy reflexivity, it must hold that \( m \sim m \).
Here, the relation \( R \) is defined such that \( m \sim_{R} m \) if \( m - m \) is even. Since \( m - m = 0 \) and 0 is indeed an even number, every integer relates to itself in this context.
Thus, the concept of reflexivity in the equivalence relation is automatically satisfied in this manner.

• Reflexivity means each element is connected to itself.
• In this case, the expression \( m - m \) evaluates to 0, an even number.
• The reflexivity condition holds true for all integers.

Defining Symmetry in Relations

The next critical property is symmetry. Symmetry in a relation means that if one element is related to another, then the reverse must also be true.
For the relation \( R \), we say \( m \sim_{R} n \) holds if \( m - n \) is even.
Suppose \( m \sim n \). By definition, this implies \( m - n = 2k \) for some integer \( k \), because an even number can be expressed as two times another integer.
We now have to show that \( n \sim m \) is also true. Consider \( n - m = - (m - n) = -2k \), and since \(-2k\) remains an even number (because any integer multiplied by 2 is even), symmetry is satisfied.

• Symmetry checks that if \( a \) is related to \( b \), then \( b \) is related to \( a \).
• Given \( m - n = 2k \), rearranging gives \( n - m = -2k\).
• The even nature remains, validating the symmetry of the relation.

Exploring Transitivity of Relations

The final aspect to confirm an equivalence relation is transitivity, which checks if an element related to a second element, and the second related to a third, implies a direct relation between the first and third elements.
For our set \( R \), if \( m \sim n \) and \( n \sim p \), this means \( m - n = 2k \) and \( n - p = 2l \) for integers \( k \) and \( l \).
Adding these gives us \( m - p = (m - n) + (n - p) = 2k + 2l = 2(k + l) \), which is still an even number.
Thus, \( m \sim p \) holds, showing transitivity aligns with our conditions for an equivalence relation.

• Transitivity ensures a continuous chain of relationships.
• Combine the conditions: \( m - n = 2k \) and \( n - p = 2l \).
• Result: \( m - p = 2(k + l) \), proving the transitivity.