Equivalence and Ordering relation

There are two classes of relations we are going to cover in the coming posts namely: equivalence relations and ordering relations. These relations have salient properties we need to be familiar:

1. Equivalence relation:

• reflexivity
• symmetry
• transitivity

1. Ordering relation:

• reflexivity
• anti-symmetry
• transitivity

There are a total of 5 properties that we have named so far. We will discuss them thoroughly, but first, let us introduce the formal definitions.

Reflexive and Irreflexive

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The digraph of a relation that is reflexive will have little loops at each element, while for irreflexive no loops will be found. Note, however, that they are not a negation of one another. Reflexive and irreflexive properties are defined only on a single quantified variable.

Reflexive Relation	Irreflexive Relation

Symmetric and Anti-symmetric

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Unlike reflexive and irreflexive relation, symmetric and anti-symmetric relations require two universally quantified variables. Symmetric and anti-symmetric relations deal with the one-way or no one-way connection of the variables.

Symmetric Relation	Anti-symmetric Relation

Transitivity

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Transitivity is a useful property as this property is required for both equivalence and ordering relations. It speaks as, “Two things that are equal to a third, are equal to each other.” Another way of stating transitivity is to say that if there’s a connection from a to b and from b to c, then there must be a connection from a to c.

_{Disclaimer: this is a summary of section 6.2 from the book A Gentle Introduction to the Art of Mathematics: by Joe Fields, the content apart from rephrasing is identical, most of the equations are screenshots of the book and the same examples are treated.}

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1. A Gentle Introduction to the Art of Mathematics by Joe Field

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Thank you for reading ...