Let be the relation on defined by . Find the domain and range of .
Domain of R = Z and Range of R = Z.
Given relation R is defined on Z (set of integers) as R = {(a, b) : a, b ∈ Z, a - b is an integer}.
For any a, b ∈ Z, the difference a - b is always an integer (since difference of two integers is always an integer). Therefore, the condition "a - b is an integer" is automatically satisfied for all a, b ∈ Z.
This means every ordered pair (a, b) where a, b ∈ Z belongs to R. Hence, R = Z × Z.
Domain: The set of all first elements of ordered pairs in R. Since every integer appears as the first element, Domain = Z.
Range: The set of all second elements of ordered pairs in R. Since every integer appears as the second element, Range = Z.
Explanation
This question tests understanding of domain and range of a relation. The key insight is recognizing that when both a and b are integers, their difference a - b is automatically an integer. This makes the relation R equal to Z × Z (all possible ordered pairs of integers). The domain (set of first elements) and range (set of second elements) are both the entire set Z. The textbook context provides the definitions: Definition 3 states domain is 'the set of all first elements' and Definition 4 discusses second elements, while Example 8 demonstrates finding domain and range from ordered pairs.
Solution Steps
Step 1: Identify that R is a relation on Z (set of integers).
Step 2: Recognize that for any a, b ∈ Z, the difference a - b is always an integer.
Step 3: Conclude that R = Z × Z (all ordered pairs of integers).
Step 4: Apply Definition 3: Domain = set of all first elements = Z.
Step 5: Apply the concept of range: Range = set of all second elements = Z.