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These two paragraphs are from wikipedia,

"In mathematics, a relation is used to describe certain properties of things. That way, certain things may be connected in some way; this is called a relation. It is clear, that things are either related, or they are not, there is no in-between. Relations are classified into four types based on mapping of elements."

"Formally, a relation is a set of n-tuples of equal degree. Thus a binary relation is a set of pairs, a ternary relation a set of triples, and so forth. In the language of set theory, a relation between two sets is a subset of their Cartesian product."

I understand the formal definition but i want to get an intuitive understanding of the concept so i want to know that in first paragraph what did author mean by saying "properties" ? I mean what kind of properties whose properties ?

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    I think the first paragraph is a little unclear. A great example of a relation on the integers is "divides". – 2017-02-25

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Properties are just anything about an object of which you can say that it is right or wrong. For example, "it is red" is a property that can apply e.g. top an apple: The apple may or may not be red.

Relations are properties that involve two or more objects. For example, "this apple is larger than that one over there." Here"is larger than" is the relation, applied to the objects "this apple" and "that one over there". It is clear that either this apple is larger than that one over there, or it is not.

So for a relation you have several sets of things your relation is meant to describe (which might be the same, as in the case of the apples above, or it may be different, e.g. for the relation "[software] runs on [hardware]", one set is the set of all software, and the other set is the set of all hardware).

A relation can be completely specified by specifying every tuple of objects for which the relation is true. For example, the ternary relation "[application] tuns under [operating system] on [hardware]" can be specified by a list of the form "Word runs under Windows on PCs, Word runs under OS X on Macs, bash runs under Linux on PCs, bash runs under FreeBSD on PCs, bash runs under Linux on Raspberry Pi, …), or in short, {(Word,Windows,PC),(Word, OS X, Mac), (bash,Linux,PC), (bash,FreeBSD,PC), (bash,Linux,Raspberry Pi), …}.

Of course in mathematics we are usually not interested in apples of software comaptibility, but in relations of mathematical objects. For example, given the set of natural numbers, you are interested in the relation "is the same number" (in short, "equals"), given by the set $\{(0,0),(1,1),(2,2),\ldots\}$. Or you might be interested in the relation "is smaller than", given by the set $\{(0,1),(0,2),(1,2),(0,3),(1,3),(2,3),(0,4),\ldots\}$. Or in the ternary relation "the sum of [number] and [number] is [number]", given by the set $\{(0,0,0),(1,0,1),(0,1,1),(2,0,2),(1,1,2),(0,2,2),\ldots\}$.

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    Thanks for the answer and very good one .It means that equality,larger than,congruence,etc are all relations . – 2017-02-25
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    @Remy: Yes. Actually, almost everything you talk about in mathematics are relations, or the objects that are related by them. – 2017-02-25
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If you are looking for some examples, here are they (the most classical ones, though) :

  • equality (in any set)

  • comparison (large) between real numbers

  • comparison (strict) between real numbers

  • divisibility in the set on positive integers

  • congruence modulo $n$ (given $n\ge1$) in set set of integers

  • inclusion in the set of all subsets of some set

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    For a non-binary example, you have the classical ternary relation in geometry given by whether three points lie on a line. Or a bit more restrictive, whether one point is between the two others. – 2017-02-25
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    @Adren Is equality a relation ? – 2017-02-25
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    @Remy: Of course it, but in some sense a trivial one since each element is only in relation with itself. – 2017-02-25