On A = {1,2,3,4,5}, R = {(1,2), (2,3), (4,5)}. How many ordered pairs are in the smallest equivalence relation containing R?
Answer and explanation
Correct answer: 13
The smallest equivalence relation containing R must be reflexive, symmetric, and transitive. The pairs (1,2) and (2,3), together with these three properties, place 1, 2, and 3 in one equivalence class. Once a class has three elements, it must contain every ordered pair among them, contributing 3² = 9 pairs, including diagonal and reverse pairs. The pair (4,5) creates a separate class {4,5}, which contributes 2² = 4 pairs. Element groups cannot be connected without adding unnecessary pairs, so the two classes remain separate in the smallest relation. Hence the total is 9 + 4 = 13, making option C correct. The values 9 and 11 omit required pairs, while 25 treats all five elements as one class and is not minimal.
Frequently asked questions
What is the correct answer to this question?
13
Why is this the correct answer?
The smallest equivalence relation containing R must be reflexive, symmetric, and transitive. The pairs (1,2) and (2,3), together with these three properties, place 1, 2, and 3 in one equivalence class. Once a class has three elements, it must contain every ordered pair among them, contributing 3² = 9 pairs, including diagonal and reverse pairs. The pair (4,5) creates a separate class {4,5}, which contributes 2² = 4 pairs. Element groups cannot be connected without adding unnecessary pairs, so the two classes remain separate in the smallest relation. Hence the total is 9 + 4 = 13, making option C correct. The values 9 and 11 omit required pairs, while 25 treats all five elements as one class and is not minimal.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Equivalence relation.
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