Equivalence class what is

It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. What exactly are equivalence classes? Equivalence classes are sets of elements which are all equivalent between them. In the same way, if the equivalence relation is "being born the same year", then each year yields a different equivalence class of all the people from this year. To sum up, an equivalence relation cuts the universe into "potatoes" of elements: inside a potato, all elements are equivalent to each other, and a potato is called an equivalence class.

Extra bonus question: Write down the two equivalence classes. Let the ground set be people, and say that two people are equivalent if they have the same type of sex organs.

Then, every man is equivalent to every other man. The set of all men is an equivalence class, as is the set of all women. Let the ground set be integers, and say that two integers are equivalent if their difference is a multiple of 2.

Every even integer is equivalent to every other even integer since the difference of two evens is even. Every odd integer is equivalent to every other odd integer similarly. Hence the two equivalence classes are the set of all even integers, and the set of all odd integers.

You have different types of clothes T-shirt, shoes, trousers If each kind of them are relationed with others clothings of the same type.

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Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. There is a close relation between partitions and equivalence classes since the equivalence classes of an equivalence relation form a partition of the underlying set, as will be proven in Theorem 7.

The proof of this theorem relies on the results in Theorem 7. We will use Theorem 7. Part 1 of Theorem 7. That is, we need to show that any two equivalence classes are either equal or are disjoint. However, this is exactly the result in Part 3 of Theorem 7. Note : Theorem 7. This process can be reversed. This will be explored in Exercise Is this set equal to any of the previous sets we have studied in this part? The Definition of an Equivalence Class We have indicated that an equivalence relation on a set is a relation with a certain combination of properties reflexive, symmetric, and transitive that allow us to sort the elements of the set into certain classes.

Progress Check 7. Answer Add texts here. Do not delete this text first. Determine the equivalence class of 0. Corollary 7. Exercise 7. Determine all of the congruence classes for the relation of congruence modulo 5 on the set of integers. In Progress Check 7.

Also, see Exercise 9 in Section 7. See Exercise 13 in Section 7. Determine all the distinct equivalence classes for this equivalence relation. In Exercise 15 of Section 7.