The relation R defined in $A=\{1,2,3\}$ by aRb, if $|a^{2}-b^{2}|\le5$, which of the following is not true?

Question

Accepted Answer

To determine why Option 4 is the correct answer, we need to analyze the relation $ R $ defined on the set $ A = \{1, 2, 3\} $ by the condition $ |a^2 - b^2| \leq 5 $.

### Explanation for Option 4:

**Option 4: Range of $ R = \{5\} $**

The range of a relation $ R $ is the set of all second elements $ b $ in the ordered pairs $ (a, b) $ that belong to $ R $. In this context, the range should be the set of all elements $ b $ such that there exists an $ a $ in $ A $ with $ |a^2 - b^2| \leq 5 $.

Let's evaluate the pairs:

- For $ a = 1 $: 
  - $ |1^2 - 1^2| = 0 \leq 5 $ 
  - $ |1^2 - 2^2| = 3 \leq 5 $
  - $ |1^2 - 3^2| = 8 \not\leq 5 $

- For $ a = 2 $:
  - $ |2^2 - 1^2| = 3 \leq 5 $
  - $ |2^2 - 2^2| = 0 \leq 5 $
  - $ |2^2 - 3^2| = 5 \leq 5 $

- For $ a = 3 $:
  - $ |3^2 - 1^2| = 8 \not\leq 5 $
  - $ |3^2 - 2^2| = 5 \leq 5 $
  - $ |3^2 - 3^2| = 0 \leq 5 $

From this analysis, the range of $ R $ is actually $ \{1, 2, 3\} $, not $\{5\}$. Therefore, Option 4 is indeed not true.

### Explanation for Other Options:

**Option 1: $ R = \{(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (3,2)\} $**

The pairs listed in Option 1 are consistent with the condition $ |a^2 - b^2| \leq 5 $. Each pair satisfies the inequality, confirming that Option 1 is true.

**Option 2: $ R^{-1} = R $**

The inverse relation $ R^{-1} $ consists of all pairs $ (b, a) $ such that $ (a, b) $ is in $ R $. Since $ R $ is symmetric (i.e., if $ (a, b) \in R $, then $ (b, a) \in R $), $ R^{-1} = R $. Therefore, Option 2 is true.

**Option 3: Domain of $ R = \{1, 2, 3\} $**

The domain of $ R $ is the set of all first elements $ a $ in the ordered pairs $ (a, b) $ that belong to $ R $. Since every element of $ A $ appears as the first element in at least one pair in $ R $, the domain is indeed $ \{1, 2, 3\} $. Thus, Option 3 is true.

In conclusion, Option 4 is the only statement that is not true, as the range of $ R $ is not $\{5\}$.