Let R be the relation on Z defined by R = { ( a , b ) } : € Z , a - b is an integer. ) Find the domain and range of R ?

Question

Answer

The relation R on Z is defined by

R={(a,b):a,b∈Z,a−b is an integer}

Since a−b is an integer for all a, b∈Z.

So, Domain (R)=Z, Range (R)=Z.

Answer

Note that [.] is greatest integer function and {.} is fractional part function. We know that for any number,

x = [x] + {x}

=> x - {x} = [x]

Since [x] is always an integer x - {x} will always be an integer.

If a = x is any real number then we must define b = {x} for a - b to be an integer.

So, the domain of (a,b) is (x,{x}) and is range is Z (all integers).

Answer

Domain and Range of a Relation

The domain and range of a relation are important concepts in mathematics that help us understand the set of input values and output values of the relation, respectively. In this case, we are given a relation R defined on the set of integers Z, where R consists of ordered pairs (a, b) such that a - b is an integer. Let's find the domain and range of this relation.

Domain of Relation R

The domain of a relation is the set of all input values for which the relation is defined. In this case, the input values are the integers a and b. Since the relation R is defined on the set of integers Z, the domain of R is also the set of integers Z.

Range of Relation R

The range of a relation is the set of all output values that the relation produces. In this case, the output values are the differences a - b, which should be an integer. Let's consider some examples to understand the range of R.

1. If we take a = 5 and b = 3, then a - b = 5 - 3 = 2, which is an integer.
2. If we take a = 8 and b = 6, then a - b = 8 - 6 = 2, which is an integer.
3. If we take a = 10 and b = 7, then a - b = 10 - 7 = 3, which is an integer.

From these examples, we can observe that the difference a - b can be any integer. Therefore, the range of R is the set of all integers Z.

Summary

To summarize, the domain of the relation R is the set of integers Z, and the range of the relation R is also the set of integers Z. This means that for any integer input values a and b, the relation R will produce an integer output value a - b. It is important to note that the domain and range of a relation can vary depending on the definition of the relation. In this case, since the relation is defined on the set of integers, the domain and range are both the set of integers.

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