Relations
Equivalence Relations and Partitions
Premium Question
Grade 11
Question:
Let $A = \{a_1, a_2, \dots, a_k\}$ be partitioned as $A = A_1 \cup A_2 \cup \dots \cup A_k$ with $A_i \cap A_j = \emptyset$ for $i \neq j$. Define $R = \{(x, y) : x, y \in A_i \text{ for some } i\}$. Then $R$ is:
(1) only reflexive
(2) only symmetric
(3) only transitive
(4) an equivalence relation
Step-by-Step Solution
Key Concept: $R$ groups elements in the same part together. Reflexive: $x \in A_i \Rightarrow (x, x) \in R\checkmark$. Symmetric: if $x, y \in A_i$ then $y, x \in A_i\checkmark$. Transitive: $x, y \in A_i$ and $y, z \in A_j$ — since $y \in A_i \cap A_j$ and parts are disjoint, $i = j$, so $x, z \in A_i\checkmark$. Equivalence.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (4)