Relations
Equivalence Relations
Premium Question
Grade 11
Question:
Let $f : A \to B$. Define $R$ on $A$ by $a \, R \, b \Leftrightarrow f(a) = f(b)$. Then $R$ is:
(1) reflexive and symmetric only
(2) an equivalence relation
(3) symmetric and transitive only
(4) reflexive and transitive only
Step-by-Step Solution
Key Concept: Reflexive: $f(a) = f(a)\checkmark$. Symmetric: $f(a) = f(b) \Rightarrow f(b) = f(a)\checkmark$. Transitive: $f(a) = f(b)$ and $f(b) = f(c) \Rightarrow f(a) = f(c)\checkmark$. This is the kernel relation of $f$; it is always an equivalence relation.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)