Relations in Mathematics have a reflexive property which basically means that every element in itself will also be related to itself. There may be other relations however the core condition to be reflexive is that it needs to align each of its elements to itself. That is basic. Others can follow after that.  Let us see an example. We take a binary relation which is termed as X and it is on a set Y and it is reflexive if it relates to every element of the set Y to itself. for example- a relation is reflexive if it is same and equal to the set of real numbers which is because we know every real number is a reflection of itself. Every possible real number is equal to itself. Thus a relation which is reflexive is the property by which it is a true representation of itself. When we talk of Equivalent relations thus every relation is reflexive along with being. Symmetrical also.

Examples of such Relations. These are found in daily mathematics and easier to grasp.

  1. Relation which is Irreflexive- This just means that it does not relate to any element to itself. IT is bigger or larger than the relation on the real numbers. Therefore every relation which we take is either reflexive or irreflexive. We can prove that some relations which are there are totally related to themselves while others are not at all. When we take even numbers then we know that the set of even numbers is reflexive and it is however compared to the set of odd numbers which is not.

Let us take an example of a Quasi Reflexive Relation – They have the same limit for the real number sequence. We need to note that not every sequence will have a limit and then we need to state that this is not reflexive. However if we take a sequence which is having the same limit and the other sequence then it has the same limit as it had for itself. Let us take an example for a Quasi reflexive relation for the left is the left EUCLIDEAN relation and this is always left quasi reflexive. This is only true as the left Quasi reflexive but not necessarily any Quasi reflexive. Quasi Reflexive is that If every element that is related to some element is also related to itself.

A relation is said to thus Reflexive if it relates each of its elements to itself. It may relate to different things but centre core is the concept that every element in itself will relate to itself. When we talk of Equal relation there is a concept that it is a reflexive relation, The symbols or operations like = or greater and equal to etc will be always reflexive and they are deemed of great use.

Let us understand what a coreflexive relation is- this is the relation which acts on integers and here each o the odd numbers is thus related to itself and there are thus no other relations. Thus the relation which is equality relation is the example of both reflexive and coreflexive relation. Thus any coreflexive relation is a subset of the identity relation.

Examples of reflexive relations are: Equality or is equal to sign. also we can include here others like is greater than or is equal to, is less than or equal to, is divisible by , is a subset of.

Examples of Irreflexive relation are: Is not owl to, is greater than, is less than, is a proper subset of. Here it means that every element does not relate to itself and is greater than relation will indicate it works on real numbers. But we also need to know that not very element which is not reflexive is irreflexive. We can also define some elements where they are related to themselves but there are others which are not.

Example of Quasi Reflexive Relation: This has the same limit on the set of real number sequence. If one sequence has same limit to that of another sequence then it is also limited to itself. Example is Left Euclidean Relation.

Example of Coreflexive relation is the integer relation and here the odd numbers are related to each other and there are no other substantial relations. Here the equality relation is example of both REFLEXIVE and COREFLEXIVE relation. The coreflexive relation is subset of the other relation.

Some other definitions related to Reflexive properties are-

Anti -reflexive

Quasi Reflexive

Left Quasi Reflexive

Right Quasi Reflexive

Anti symmetric


Learn more: Relations from Class 12 Maths