Answers

2014-05-20T19:36:12+05:30
Well, the collection of equivalence classes is a partition of the set. As such, two key properties have to hold: 

1) Every element in the set must appear in one of the equivalence classes, and 
2) distinct equivalence classes have to be disjoint---meaning each term appears in exactly one equivalence class. 

So, we can examine each of the sets in 1.--5. to make sure these hold. 

1. Every element a, b, c, d, and e appears. There is only one set contained in that set, so each element only appears once. That's it. It has the required properties, so it could be the set of equivalence classes for an equivalence relation. 

2. The element e doesn't appear in any of those subsets. So it can't be a partition. 

The answers for the remaining three are 3. Yes, 4. No, 5. Yes. Just use the acid test described above to justify each of these answers.
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2014-05-21T01:30:38+05:30
In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of elements that are related to one another, forming what are called equivalence classes. Notationally, given a setX and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in Xwhich are equivalent to a. It follows from the definition of the equivalence relations that the equivalence classes form a partition of X. The set of equivalence classes is sometimes called the quotient set of X by ~ and is denoted by X / ~.When X has some structure, and the equivalence relation is defined with some connection to this structure, the quotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and the quotient category.
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