Question

Which of the following statements is \textit{not true about equivalence classes \( A_i \) (\( i = 1, 2, \ldots, n \)) formed by an equivalence relation \( R \) defined on a set \( A \)?}

• A. \( \bigcup_{i=1}^n A_i = A \)
• B. \( A_i \cap A_j \neq \phi, \, i \neq j \)
• C. \( x \in A_i \text{ and } x \in A_j \implies A_i = A_j \)
• D.

Hint:

Equivalence classes are disjoint by definition. If two equivalence classes intersect, they are the same class. Always analyze their properties to identify valid and invalid statements.

Answer 1

The Correct Option is B

Step 1: Understand the properties of equivalence classes. Equivalence classes are subsets of a set \( A \) defined by an equivalence relation \( R \). The important properties of equivalence classes are: 
The union of all equivalence classes equals the set \( A \): \[ \bigcup_{i=1}^n A_i = A. \] Equivalence classes are mutually exclusive (disjoint), meaning: \[ A_i \cap A_j = \emptyset, \quad \text{for } i \neq j. \] If an element \( x \) belongs to two equivalence classes, then those two classes are identical: \[ x \in A_i \text{ and } x \in A_j \implies A_i = A_j. \] Every element within an equivalence class \( A_i \) is related to every other element in \( A_i \) under the equivalence relation \( R \). 
Step 2: Evaluate the given options. (A): True, because the union of all equivalence classes forms the set \( A \) by definition. 
(B): False, since equivalence classes are disjoint and cannot overlap. Their intersection is always empty for \( i \neq j \). 
(C): True, as elements belonging to multiple equivalence classes imply those classes are identical. 
(D): True, because all elements within the same equivalence class are related under the equivalence relation. 
Step 3: Final Answer. The statement in option (B) is {not} true.