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THE NOTATION xRy

**The Notation xRy –**

**If (x, y) is a member of a relation R,
then we also use the symbolism ‘xRy’ and read ‘x is the relation R to y’.**

**(x, y) ∈ R => xRy **

**If, (x, y) ∉ R,**

**For example, for the relation
‘is the capital of’, we have ‘Beijing R China’, ‘Colombo R Sri-Lanka’, ‘Dhaka R
Bangladesh’. We give below more examples
of relations which can be expressed via the ‘xRy’ symbolism.**

**(i) Let ‘R’ mean ‘is equal to’.
Then ‘xRy’ means x = y**

**(ii) Let A = {2, 3, 4,
5,………….,12} and R means ‘is one-third
of’ in A X A, **

**then 2R6, 3R9, 4R12 =>
R = {(2, 6), (3, 9), (4, 12)}**

__Note.1)__ We emphasize that
relations are sets and therefore any statements about sets or any operation
defined on sets are appropriately applied to relations. One can thus speak of
the intersection or union of two relations or speak of one relation being a
subset of another.

__Note.2)__ In the above picture ,
we have seen that the product set A X B is a set of ordered pairs (a ∈ A, b ∈
B). Since a relation is also a set of ordered pairs, therefore a relation is
the subset of a product set.