Mathematical Relations and Properties

A set of vocabulary flashcards covering the definitions and properties of reflexive, symmetric, transitive, and equivalence relations as discussed in the lecture.

Reflexive Relation

A relation on a set $A$ where (a, a) \text{\textin} r for every a \text{\textin} A; for example, if $A = {1, 2, 3}$, the relation contains $(1, 1)$, $(2, 2)$, and $(3, 3)$.

Symmetric Relation

A relation where if (a, b) \text{\textin} R, then it must also be true that (b, a) \text{\textin} R.

Transitive Relation

A relation where if (a, b) \text{\textin} R and (b, c) \text{\textin} R, then (a, c) \text{\textin} R must also be true.

Equivalence Relation

A relation that satisfies three specific conditions: it is Reflexive, Symmetric, and Transitive (often abbreviated as RST).

Non-Symmetric Example

In the set $A = {1, 2, 3}$, the relation $R = \text{\textbraceleft}(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\text{\textbraceright}$ is not symmetric because (1, 2) \text{\textin} R but (2, 1) \text{\textnotin} R.

Non-Transitive Example

In the set $A = {1, 2, 3}$, the relation $R = \text{\textbraceleft}(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\text{\textbraceright}$ is not transitive because while (1, 2) \text{\textin} R and (2, 3) \text{\textin} R, the pair (1, 3) \text{\textnotin} R.