Relation Property

Abstract

---

• Not properties of the elements of the Set!

Reflexive

$$\forall x\in A(xRx)$$

• Every element in the given Set must be related to itself

Symmetric

$$\forall x,y\in A(xRy\to yRx)$$

• If an element is related to another element, the another element must be related to this element too

Transitive

$$\forall x,y,z\in A(xRy\cap yRz\to xRz)$$

• If an element is related to another element, and that element is related to a third element. Then this must be related to the third element

Equivalence Relation

---

• Let $A$ be a Set and $R$ be a Relation on $A$
• $R$ is equivalence relation iff $R$ is Reflexive, Symmetric and Transitive

Equivalence Class

• Basically same as the component of a Set Partition or elements of Equivalence Relation
• Can be represented with $[a{]}_{relation}$, it means the Equivalence Class contains element $a$
• $[a{]}_{relation}$ and $[b{]}_{relation}$ are the same iff $b$ is in the same equivalence class as $a$