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Binary relation

Definition Let A and B be nonempty sets. A binary relation

from A to B is a subset of the Cartesian Prodnet A XB. A binary relation from A to A is called a binary relation on A.

Let R CA XB (R is a binary relation from A to B).

If (a, b) C /?, we say: a is related to b and denote: aRb. If (a, b) /C R, we say: a is not related to b and denote: a 6Rb.

Examples:

1.    Let A = fl, 2, 3/ and B = {a, b, c}.

Then /? = /(!, a), (1, c), (2, c)} is binary relation from A to B.

In this case we have: 1 Ra, 1 Rc, 2Rc

and also: 1 6Rb, 2 6Ra, 2 6Rb, 3 6Ra, 3 6Rb, 3 6Rc.

2.    (R, R2), defined as: Va, b CR aRb da = b.

This is the eąuality relation on R.

3.    (S,R2), defined as: Va, b CR aSbda 6 b.

This is the relation “less than or equal” on R.