# Properties of Relations

A relation R from the set A to set B is a subset of A * B. If A is any set then any subset of A * A is a relation on A. If R is relation from the set A to set B and (a, b) $\in$ R, then we say a is related to b under the relation R. This is denoted by a R b. If (a, b) not equal R then we write a R b under the relationship R.**Example:** Consider the sets A = {1, 2, 3, 4} and B = {3, 5, 7}

Set R = |(a, b)| a $\in$ A, b $\in$ B a < b}, is a relation from the set A into set B.**Solution:** If A has m elements and B has n elements, then we know that A * B has mn elements. Thus there are 2$^{mn}$ subsets of A * B. Hence there are 2$^{mn}$ different realtions from the set A into B.

Let R $\subset$ A * B be a relation from the set A into set B . Then the set

{ a | (a, b) $\in$ R}

is called the domain of the relation R and the set {a| (a, b) $\in$ R} is called the range of the relation R.