How many of the following are not properties of equivalence classes?
- All elements of an equivalence class will be related to each other
- No element of an equivalence class will be related to an element of another equivalence class
- All the equivalence classes, which are sets, are disjoint
- Union of all the equivalence classes of particular relation will give the set A on which we defined the relation
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How many of the following are not properties of equivalence classes?
- All elements of an equivalence class will be related to each other
- No element of an equivalence class will be related to an element of another equivalence class
- All the equivalence classes, which are sets, are disjoint
- Union of all the equivalence classes of particular relation will give the set A on which we defined the relation
When we define an equivalence class on a set, we get equivalence classes. An equivalence class of an element is the set of all elements related to that particular element. Since an equivalence relation is also reflexive, we can say that the element we are considering would also be related to itself. So 'a' will be part of equivalence class of a.
Now let's take any two elements, b and c, from the equivalence class of 'a'. We know that b is related to a and a is related to c. Since an equivalence relation is transitive, we can say that b is related to c.
From this, we can say that any two element in an equivalence class will be related to all the elements of that equivalence class.
Any element 'a' of an equivalence class won't be related to any element of another equivalence class because, any element related to 'a' should also be part of equivalence class of 'a'.
We saw that every element in an equivalence relation will be related to itself. We also saw that no element of an equivalence class will be related to an element of another equivalence class. Combining these two we can say that there won't be any common element between any two equivalence classes. This is same as saying all the equivalence classes are disjoint.
Since an equivalence relations divides the set to get to equivalence class, we can say that the union of all such relations will give the set on which the relation is defined. So the last sentence is also correct.
=> number of statements which are wrong = 0
When we define an equivalence class on a set, we get equivalence classes. An equivalence class of an element is the set of all elements related to that particular element. Since an equivalence relation is also reflexive, we can say that the element we are considering would also be related to itself. So 'a' will be part of equivalence class of a.
Now let's take any two elements, b and c, from the equivalence class of 'a'. We know that b is related to a and a is related to c. Since an equivalence relation is transitive, we can say that b is related to c.
From this, we can say that any two element in an equivalence class will be related to all the elements of that equivalence class.
Any element 'a' of an equivalence class won't be related to any element of another equivalence class because, any element related to 'a' should also be part of equivalence class of 'a'.
We saw that every element in an equivalence relation will be related to itself. We also saw that no element of an equivalence class will be related to an element of another equivalence class. Combining these two we can say that there won't be any common element between any two equivalence classes. This is same as saying all the equivalence classes are disjoint.
Since an equivalence relations divides the set to get to equivalence class, we can say that the union of all such relations will give the set on which the relation is defined. So the last sentence is also correct.
=> number of statements which are wrong = 0
