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Definition of Relations


Let A and B be two nonempty sets. A relation R from A to B is a subset of A × B.
If R Í A × B and (a, b) R we say that a is related to b by R and we write a R b. If a is not related to b by R we write a b.
In case A and B are the same set, i.e. if R Í A × A, we say R is a relation on A or a relation from A to A.

Illustration
Let A = {1, 2, 3} and B = {3, 5, 7}. Write the relation R from A to B such that x R y if x < y.

Solution
Since 1 A is less than 3, 5, 7 B, (1, 3), (1, 5), (1, 7) R.
Similarly, (2, 3), (2, 5), (2, 7) R.
For 3 A, as 3 = 3, (3, 3) Ï R.
But since 3 < 5, 3 < 7, (3, 5), (3, 7) R.
Hence R = {(1, 3), (1, 5), (1, 7), (2, 3), (2, 5), (2, 7), (3, 5), (3, 7)}.

Examples
(i) Let A = {1, 2, 4, 5}. Let R be a relation on A defined by x R y if x divides y.

Solution
As 1 divides every natural number, 1 R x for each x A. That is, (1, 1), (1, 2), (1, 4), (1, 5) R.
Also, as 2 divides 2 and 2R2 and 2R4. Since 2 does not divide 1 and 5, (2, 1), (2, 5) Ï R.
Similarly (4, 4) and (5, 5) R.
Thus, R = {(1, 1), (1, 2), (1, 4), (1, 5), (2, 2), (2, 4), (4, 4), (5, 5)}.

(ii) Let A = {1, 2, 4, 5, 8, 64, 625}. Define R on A by x R y if x3 = y.

Solution
(ii) Since 1 = 13, 23 = 8, 43 = 64 and 53 = 625,
R = {(1, 1), (2, 8), (4, 64), (5, 625)}.




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