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What do we mean by the word binary relation. The words" binary relation on some set say S"Is commonly used in many textbooks.what is the use of defining a binary relation on some set and try to see how the elements in the set are related. normally relation is defined to be the subset of Cartesian product of two sets say A and B.but here we are talking about one set S and not two. What is the reason for this.

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    "binary relation" means a relation between two "entities" : "$x$ is Father of $y$", "$x$ is less than $y$". – 2017-01-01
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    Our life (mathematical and not) is plenty of them : greater than, older than, richer than,... Basically, when we "compare" two things we are defining (using) a relation. – 2017-01-01
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    Consider the relation $R$ "$x$ is husband of $y$"; you can consider it as defined on the set $H$ of humans (and thus : $R \subseteq H \times H$) or on the sets $M$ of men and $W$ of women ( and thus : $R \subseteq M \times W$). – 2017-01-01

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A binary relation on a set $S$ is a subset $R$ of the Cartesian product $S\times S$. If $(a,b)\in R$, we say $a$ and $b$ are related and write $aRb$ as a shorthand. Note that this does not necessarily imply that $bRa$ holds, though when it does, we call the relation symmetric.

The reason we care about binary relations is because they relate pairs of elements. Many things are binary relations. For example, equality ($=$) of real numbers is a binary relation. Inequality ($\leq$) is another example.

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    Sir normally relation is defined to be the subset of Cartesian product of two sets say A and B.but here we are talking about one set S and not two. What is the reason for this. – 2017-01-01
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    In your question, you asked about a relation on a *single* set $S$. If you have two sets, $A$ and $B$, a relation on them is, as you say, a subset of $A\times B$. – 2017-01-01
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    Yeah what is the difference between the relation on two sets and the relation on a single set ....Please help me – 2017-01-01
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    The definition involving two sets is more general. Setting $A=B$ gives you the definition involving one set. – 2017-01-01