To prove that $R$ is an equivalence relation on the set $A = \{1, 2, 3, 4, 5\}$, we need to verify the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

Step 1: Reflexivity

A relation $R$ is reflexive if $(a, a) \in R$ for all $a \in A$.

The definition of $R$ is: $R = \{(a, b) : a - b \text{ is even}\}.$

For reflexivity:

• Let $a \in A$.
• Since $a - a = 0$, which is even, $(a, a) \in R$.
• Hence, $R$ is reflexive.

Step 2: Symmetry

A relation $R$ is symmetric if $(a, b) \in R$ implies $(b, a) \in R$ for all $a, b \in A$.

• If $(a, b) \in R$, then $a - b$ is even.
• An even number satisfies $a - b = -(b - a)$, which means $b - a$ is also even.
• Thus, $(b, a) \in R$.
• Therefore, $R$ is symmetric.

Step 3: Transitivity

A relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R$ imply $(a, c) \in R$ for all $a, b, c \in A$.

• If $(a, b) \in R$ and $(b, c) \in R$, then $a - b$ is even and $b - c$ is even.
• Adding these, $(a - b) + (b - c) = a - c$ is also even.
• Thus, $(a, c) \in R$.
• Therefore, $R$ is transitive.

Since $R$ satisfies reflexivity, symmetry, and transitivity, $R$ is an equivalence relation.

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Step 4: Equivalence Classes

The equivalence classes of $R$ group elements of $A$ that satisfy $(a, b) \in R$, i.e., $a - b$ is even. This means $a$ and $b$ must have the same parity (both even or both odd).

• Odd numbers in $A$: $\{1, 3, 5\}$ (all differences are even).
• Even numbers in $A$: $\{2, 4\}$ (all differences are even).

Thus, the equivalence classes are: [ [1] = {1, 3, 5}, \quad [2] = {2, 4}. ]

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Final Answer

1. $R$ is an equivalence relation because it satisfies reflexivity, symmetry, and transitivity.
2. The equivalence classes of $R$ are: [ [1] = {1, 3, 5}, \quad [2] = {2, 4}. ]

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Would you like a deeper explanation of any step, or clarification about equivalence classes?

Relative Questions

1. Can you explain the importance of equivalence relations in set theory?
2. What happens if the set $A$ is expanded to include negative integers?
3. How would the equivalence classes change if $R$ was defined differently?
4. Can you find the equivalence relation matrix for $R$?
5. How are equivalence classes related to partitions of a set?

Tip: When working with equivalence relations, always start by verifying the three defining properties systematically.