Question 18 of 33intermediate💡 UnderstandShort Answer2 marks

Let RR be the relation on ZZ defined by R={(a,b):a,b∈Z,a−b is an integer}R = \{(a, b) : a, b \in Z, a - b \text{ is an integer}\}. Find the domain and range of RR.

Correct Answer

The relation RR is defined on the set of integers ZZ. For any a,b∈Za, b \in Z, the difference a−ba - b is always an integer. Therefore, the condition a−b∈Za - b \in Z holds true for all ordered pairs (a,b)(a, b) where aa and bb are integers. Consequently, R=Z×ZR = Z \times Z.

According to Definition 3, the domain is the set of all first elements of the ordered pairs. Since the first element aa can be any integer in ZZ, the domain of RR is ZZ.

Similarly, the range is the set of all second elements of the ordered pairs (as illustrated in Example 8). Since the second element bb can be any integer in ZZ, the range of RR is ZZ.

Exercise: EXERCISE 2.2 | Q: 9 | (Page 7)
For More Understanding

Explanation

The question asks for the domain and range of a relation on ZZ. The context provides Definition 3, which states the domain is the set of first elements. While Definition 4 in the text contains a likely typo (calling the set of second elements 'codomain'), Example 8 clarifies that the set of second elements is the 'range'. Since integers are closed under subtraction, a−ba-b is always an integer for any a,b∈Za,b \in Z. Thus, all pairs (a,b)(a,b) are valid, making the domain and range both equal to the set ZZ.

Solution Steps

  1. Step 1: Analyze the relation R={(a,b):a,b∈Z,a−b is an integer}R = \{(a, b) : a, b \in Z, a - b \text{ is an integer}\}.

  2. Step 2: Recognize that for any integers aa and bb, the difference a−ba - b is always an integer. Thus, the condition holds for all a,b∈Za, b \in Z.

  3. Step 3: Conclude that RR contains all ordered pairs in Z×ZZ \times Z.

  4. Step 4: Apply Definition 3 to find the domain: the set of first elements is ZZ.

  5. Step 5: Apply the logic from Example 8 to find the range: the set of second elements is ZZ.

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