Question

What is a reflexive property? :/ Please help!

Answer 1

A reflexive property is a property of a relation on a set such that for every member, a, of the set, a is in the given relation to itself. For example, take the set of integers, then the ordinary relation "=" (equals) has the reflexive property, since for every integer, n, we have n=n.

Another example on the set of integers would be the relation "|" (divides w/o a remainder), since for every integer n, we have n|n.

Question for you: what prompted your question? Depending on the underlying set, there are potentially other examples that could be constructed, not all from the field of mathematics or abstract logic.

Answer 2

Here is kind of a 10,00 foot view; I hope is not too broad.

An example where reflexive properties are used is in defining an equivalence relation; such a relation also defines a set partition. I always found it fascinating that the way to define similarity or equivalence ("all of THESE things are the same") also determines difference ("all of THESE things are different from all of THOSE things"). This example shows how the reflexive property is used or shows up in 'rel life' so to speak.

The 12 zodiac signs define 12 equivalence classes of people (all the capricorn people, all the gemini people, etc.). Each person belongs to exactly one class or zodiac sign. All the people in a given class (all the capricorns, say) relate to each other with a relationship (call it R) which means "has the same sign as". That relation R has three properties, and one of those properties is the reflexive property.

"I have the same astrological sign as myself" would be a statement of the reflexive property of this particular relation. To define a class of "similars" (an equivalence class) it is necessary that each member of the class relate to itself, and so the relation necessarily exhibits the reflexive property.