The basic nature of the hydroxides of group `13` decreases progressively down the group.(T/F)

Explanation

To determine whether the statement "The basic nature of the hydroxides of group 13 decreases progressively down the group" is true or false, we can analyze the hydroxides of the elements in group 13 step by step.

### Step 1: Identify the elements in Group 13
Group 13 of the periodic table consists of the following elements:
- Boron (B)
- Aluminum (Al)
- Gallium (Ga)
- Indium (In)
- Thallium (Tl)

### Step 2: Write the hydroxides of these elements
The hydroxides corresponding to these elements are:
- Boron hydroxide: \( B(OH)_3 \) (Boric acid)
- Aluminum hydroxide: \( Al(OH)_3 \)
- Gallium hydroxide: \( Ga(OH)_3 \)
- Indium hydroxide: \( In(OH)_3 \)
- Thallium hydroxide: \( Tl(OH)_3 \)

### Step 3: Analyze the nature of these hydroxides
- **Boron hydroxide** is known to be acidic in nature.
- **Aluminum hydroxide** is amphoteric, meaning it can act as both an acid and a base.
- **Gallium hydroxide**, **Indium hydroxide**, and **Thallium hydroxide** are basic in nature.

### Step 4: Understand the trend in basicity
As we move down the group from Boron to Thallium, the size of the cation increases:
- \( B^{3+} \) (smallest)
- \( Al^{3+} \)
- \( Ga^{3+} \)
- \( In^{3+} \)
- \( Tl^{3+} \) (largest)

### Step 5: Relate cation size to basicity
The basic nature of hydroxides is related to the ability of the cation to hold onto the hydroxide ion (\( OH^- \)):
- As the size of the cation increases, the attraction between the cation and the \( OH^- \) ion decreases.
- This decrease in attraction allows the \( OH^- \) ions to be released more easily, which increases the basicity of the hydroxide.

### Step 6: Conclusion
Based on the analysis:
- The basic nature of the hydroxides does not decrease; it actually increases as we move down the group from Boron to Thallium.
- Therefore, the statement "The basic nature of the hydroxides of group 13 decreases progressively down the group" is **False**.

### Final Answer
**False**