Given that[HS 12]।R = {(a, b)| 3 divides a - b}is an equivalence relation in the set of integers Z.What is the number of partitions of Z ? (অখণ্ড ।সংখ্যাৰ সংহতি Z অত R = {(a,b)| 3 ৰে বিভাজ্যa – b} এটা সমতুল্য সম্বন্ধ। Zঅৰ বিভাগৰ সংখ্যা কিমান ?)[FIS '13]
Answers
Step-by-step explanation:
Given as R = {(a, b): a – b is divisible by 3; a, b ∈ Z} is a relation To prove equivalence relation, the given relation should be reflexive, symmetric and transitive. We have to check these properties on R. Reflexivity: Let a be an arbitrary element of R. Then, a – a = 0 = 0 × 3 ⇒ a − a is divisible by 3 ⇒ (a, a) ∈ R for all a ∈ Z Therefore, R is reflexive on Z. Symmetry: Let (a, b) ∈ R ⇒ a − b is divisible by 3 ⇒ a − b = 3p for some p ∈ Z ⇒ b − a = 3 (−p) Here, −p ∈ Z ⇒ b − a is divisible by 3 ⇒ (b, a) ∈ R for all a, b ∈ Z Clearly, R is symmetric on Z. Transitivity: Let (a, b) and (b, c) ∈ R ⇒ a − b and b − c are divisible by 3 ⇒ a – b = 3p for some p ∈ Z And b − c = 3q for some q ∈ Z On adding above two equations, we get a − b + b – c = 3p + 3q ⇒ a − c = 3(p + q) Here, p + q ∈ Z ⇒ a − c is divisible by 3 ⇒ (a, c) ∈ R for all a, c ∈ Z Thus, R is transitive on Z. ∴ R is reflexive, symmetric and transitive. Clearly, R is an equivalence relation on Z.
Answer:
the correct answer is {.........,-4,-1,2,5,8........}