Question 11 of 33beginner💡 UnderstandShort Answer2 marks

Let A={1,2,3,…,14}A = \{1, 2, 3, \ldots, 14\}. Define a relation RR from AA to AA by R={(x,y):3x−y=0,where x,y∈A}R = \{(x, y): 3x - y = 0, \text{where } x, y \in A\}. Write down its domain, codomain and range.

Correct Answer

Domain = {1, 2, 3, 4}

Codomain = {1, 2, 3, ..., 14}

Range = {3, 6, 9, 12}

Exercise: EXERCISE 2.2 | Q: 1 | (Page 6)
For More Understanding

Explanation

This question asks students to find the domain, codomain, and range of a relation defined by the equation 3x - y = 0. The key insight is recognizing that this equation can be rewritten as y = 3x, meaning for each x value in set A, y must equal 3x AND y must also belong to set A. Students must systematically check which x values produce y values that remain within the bounds of set A (1 to 14). Only x values 1, 2, 3, and 4 satisfy this condition, as x = 5 would give y = 15, which exceeds the upper bound of A.

Solution Steps

  1. Step 1: Rewrite the relation condition 3x - y = 0 as y = 3x.

  2. Step 2: Find all ordered pairs (x, y) where x ∈ A and y = 3x ∈ A. For x = 1: y = 3 ∈ A ✓. For x = 2: y = 6 ∈ A ✓. For x = 3: y = 9 ∈ A ✓. For x = 4: y = 12 ∈ A ✓. For x = 5: y = 15 ∉ A ✗. For x > 5: y > 15 ∉ A ✗.

  3. Step 3: Write R in roster form: R = {(1, 3), (2, 6), (3, 9), (4, 12)}.

  4. Step 4: Domain = Set of first elements = {1, 2, 3, 4}.

  5. Step 5: Codomain = Set A = {1, 2, 3, ..., 14}.

  6. Step 6: Range = Set of second elements = {3, 6, 9, 12}.

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