Let R be a relation from N to N defined by R = \{(x, y): x, y \in N \text{ and } x^{2}=y^{3}\}. Which of the following are <strong>NOT</strong> correct? 1. (x,x)\in R for all x\in N 2. (x,y)\in R\Rightarrow(y,x)\in R 3. (x,y)\in R and (y,z)\in R\Rightarrow(x,z)\in R Select the correct answer using the code given below :
- A. 1 and 2 only
- B. 2 and 3 only
- C. 1 and 3 only
- D. 1, 2 and 3 ✓
Correct Answer: D. 1, 2 and 3
Explanation
Statement 1: For reflexive, x^2 = x^3 must hold for all x \in N, which is false (only holds for x=1). Statement 2: For symmetric, x^2 = y^3 does not imply y^2 = x^3 (e.g., 8^2 = 4^3 holds, but 4^2 = 8^3 is false). Statement 3: For transitive, x^2 = y^3 and y^2 = z^3 implies y^6 = x^4 = z^9, giving x^4 = z^9, which does not guarantee x^2 = z^3. So, all three statements are incorrect.
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