Types of Relations

Let A be a non-empty set. A binary relation R on A is said to be:

1. REFLEXIVE, if $(a,a)\in R$ for EVERY $a\in A$ — i.e. $aRa$ for every $a\in A$ (every element relates to itself).
2. SYMMETRIC, if $(a,b)\in R \Rightarrow (b,a)\in R$ for all $a,b\in A$ — i.e. whenever $aRb$, it must also be true that $bRa$ (the symbol '⇒' is read as 'implies').
3. TRANSITIVE, if $(a,b)\in R$ and $(b,c)\in R \Rightarrow (a,c)\in R$ for all $a,b,c\in A$ — i.e. whenever $aRb$ and $bRc$ both hold, it must also be true that $aRc$. EQUIVALENCE RELATION: a relation which is reflexive, symmetric, AND transitive (all three together) is called an equivalence relation. ILLUSTRATIVE EXAMPLES: (1) Let $R=\{(a,b)/a,b\in Q \text{ and } a-b\in Z\}$ on the rationals Q. Reflexive: $a-a=0\in Z$, so $(a,a)\in R$ — reflexive. Symmetric: if $(a-b)\in Z$ then $-(a-b)=(b-a)\in Z$ too, so $(a,b)\in R \Rightarrow (b,a)\in R$ — symmetric. Transitive: if $(a-b)\in Z$ and $(b-c)\in Z$, their sum $(a-b)+(b-c)=(a-c)\in Z$ — transitive. So R is an equivalence relation. (2) The relation 'is congruent to' on the set of all triangles in a plane is an equivalence relation: reflexive since $\Delta\cong\Delta$ for every triangle; symmetric since $\Delta_1\cong\Delta_2 \Rightarrow \Delta_2\cong\Delta_1$; transitive since $\Delta_1\cong\Delta_2$ and $\Delta_2\cong\Delta_3$ together give $\Delta_1\cong\Delta_3$. …