Question

Let A={0, 3, 4, 6, 7, 8, 9, 10 } and R be the relation defined on A such that R = {(x, y)∈A×A:x-y is odd positive integer or x-y=2}. The minimum number of elements that must be added to the relation R, so that it is a symmetric relation, is equal to _______.

Hint:

To make a relation symmetric, ensure for every \((x, y)\), the pair \((y, x)\) exists.

Answer 1

Correct Answer: 19

Step 1: Define the relation.
- \(A = \{0, 3, 4, 6, 7, 8, 9, 10\}\).
- \(R = \{(x, y) : x - y \text{ is odd positive integer or } x - y = 2\}\). 
Step 2: Check for symmetry.
- For each pair \((x, y) \in R\), ensure \((y, x) \in R\) to make \(R\) symmetric. 
Step 3: Count the missing pairs.
- Add 15 pairs for odd positive differences and 4 pairs for \(x - y = 2\). 
Final Answer: A minimum of 19 pairs must be added.