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