# equivalence relation examples

The last examples above illustrate a very important property of equivalence classes, namely that an equivalence class may have many di erent names. Then Ris symmetric and transitive. Let . The intersection of two equivalence relations on a nonempty set A is an equivalence relation. The quotient remainder theorem. But di erent ordered … We say is equal to modulo if is a multiple of , i.e. Equality modulo is an equivalence relation. Modulo Challenge (Addition and Subtraction) Modular multiplication. Then is an equivalence relation. A rational number is the same thing as a fraction a=b, a;b2Z and b6= 0, and hence speci ed by the pair ( a;b) 2 Z (Zf 0g). We write X= ˘= f[x] ˘jx 2Xg. An equivalence relation is a relation that is reflexive, symmetric, and transitive. Example 5: Is the relation $\geq$ on $\mathbf{R}$ an equivalence relation? Theorem. The following generalizes the previous example : Definition. For example, take a look at numbers $4$ and $1$; $4 \geq 1$ does not imply that $1 \geq 4$. Examples of Reflexive, Symmetric, and Transitive Equivalence Properties An Equivalence Relationship always satisfies three conditions: (For organizational purposes, it may be helpful to write the relations as subsets of A A.) If two elements are related by some equivalence relation, we will say that they are equivalent (under that relation). Modular addition and subtraction. Let ˘be an equivalence relation on X. Solution: Relation $\geq$ is reflexive and transitive, but it is not symmetric. Proof. This is true. In the above example, for instance, the class of … What about the relation ?For no real number x is it true that , so reflexivity never holds.. Example 6. If x and y are real numbers and , it is false that .For example, is true, but is false. First we'll show that equality modulo is reflexive. Conversely, any partition induces an equivalence relation.Equivalence relations are important, because often the set S can be ’transformed’ into another set (quotient space) by considering each equivalence class as a single unit. An example from algebra: modular arithmetic. For example, if [a] = [2] and [b] = [3], then [2] [3] = [2 3] = [6] = [0]: 2.List all the possible equivalence relations on the set A = fa;bg. It provides a formal way for specifying whether or not two quantities are the same with respect to a given setting or an attribute. The equivalence relation is a key mathematical concept that generalizes the notion of equality. An equivalence relation on a set induces a partition on it. Example. Equivalence relations A motivating example for equivalence relations is the problem of con-structing the rational numbers. Practice: Modular multiplication. This is the currently selected item. Problem 2. Answer: Thinking of an equivalence relation R on A as a subset of A A, the fact that R is re exive means that This is false. Problem 3. Equivalence relations. If we consider the equivalence relation as de ned in Example 5, we have two equiva-lence … Let Rbe a relation de ned on the set Z by aRbif a6= b. The set [x] ˘as de ned in the proof of Theorem 1 is called the equivalence class, or simply class of x under ˘. Practice: Modular addition. Examples of Equivalence Relations. Equality Relation It was a homework problem. It is true that if and , then .Thus, is transitive. The relation is symmetric but not transitive. De nition 4. Modular exponentiation. We have already seen that $$=$$ and $$\equiv(\text{mod }k)$$ are equivalence relations. Some more examples… Show that the less-than relation on the set of real numbers is not an equivalence relation. Proof. Let be an integer. if there is with . Proof. Two elements are related by some equivalence relation be helpful to write the as. 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