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Question

In a class of 50 students, 30 students play cricket and 30 students play football and everyone plays at least one of these sports and no one plays any other sport. A relation is defined on the set of students such that aRb if ‘a’ and ‘b’ play a same sport. How many equivalence classes will be formed by this relation

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Solution

Equivalence classes are formed when we define an equivalence relation on a given set. We are given a relation, which may or may not be equivalence. We will check first if the given relation is equivalence as it is not given in the question and "none of these" is one of the answer option.

We know that for a relation to be equivalence it has to be reflexive, symmetric and transitive. If ‘x’ and ‘y’ plays the same sport and ‘y’ and ‘z’ plays the same sport, can we say that x and z play the same sport? Not necessarily, right? This is true only if there are no students who play both the games. But in this case there are definitely some students who play both the games, because the sum of students who play cricket and football exceeds the total number of students in the class. So, we can say that the given relation is not transitive, hence not equivalence also. This means that no equivalence class will be formed, simply because the given relation not an equivalence relation.

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