Prove that the relation R in the set $A = {1, 2, 3, 4, 5}$ given by $R = {(a, b) : |a - b| is even}$ is an equivalence relation.

Step 1: Understand Equivalence Relation

An equivalence relation partitions a set into disjoint subsets where all elements within a subset are related to each other. To prove that a given relation is an equivalence relation, we must demonstrate that it satisfies three fundamental properties: reflexivity, symmetry, and transitivity.

Step 2: Check for Reflexivity

For a relation to be reflexive, every element must be related to itself. In this case, we need to check if $|a - a|$ is even for any element $a$ in the set $A$. The absolute difference between an element and itself is always 0.

Step 3: Evaluate Reflexivity Condition

Since 0 is an even number (as it can be written as $2 \times 0$), the condition $|a - a|$ is even is satisfied for all $a \in A$. Therefore, the relation R is reflexive.

Step 4: Check for Symmetry

For symmetry, if an element $a$ is related to an element $b$, then $b$ must also be related to $a$. We assume that $(a, b) \in R$, which means that the absolute difference between $a$ and $b$ is an even number.

Step 5: Evaluate Symmetry Condition

The absolute value of a number is the same as the absolute value of its negative. Therefore, $|a - b|$ is equal to $|b - a|$. Since we assumed $|a - b|$ is even, it follows that $|b - a|$ must also be even. This implies that $(b, a) \in R$, so the relation R is symmetric.

Step 6: Check for Transitivity

For transitivity, if $a$ is related to $b$, and $b$ is related to $c$, then $a$ must be related to $c$. We assume that $(a, b) \in R$ and $(b, c) \in R$. This means that $a - b$ and $b - c$ are both even integers (positive or negative, as their absolute values are even).

Step 7: Express differences as even integers

If the absolute value of a difference is even, then the difference itself must be an even integer (it could be positive or negative). So, $a-b$ can be written as an even integer $2m$ and $b-c$ can be written as an even integer $2n$, where $m$ and $n$ are integers.

Step 8: Find the difference between a and c

To check if $(a, c) \in R$, we need to evaluate the difference $a - c$. We can express $a - c$ as a sum of the differences $a - b$ and $b - c$, which we already know are even integers.

Step 9: Substitute and sum the even integers

Substituting the representations for $a - b$ and $b - c$ as $2m$ and $2n$ respectively, we find that their sum $2m + 2n$ simplifies to $2(m + n)$. Since $m$ and $n$ are integers, their sum $(m + n)$ is also an integer. This means that $a - c$ is twice an integer.

Step 10: Evaluate Transitivity Condition

Since $a - c$ can be expressed as $2 \times \text{an integer}$, it is an even integer. Therefore, its absolute value $|a - c|$ is also even. This proves that $(a, c) \in R$, and thus the relation R is transitive.

Step 11: Conclusion

Since the relation R satisfies all three properties required for an equivalence relation (reflexivity, symmetry, and transitivity), we can conclude that R is an equivalence relation on the set A.

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