Given universal set = `{ -6 , -5 (3)/(4) , -sqrt(4), - (3)/(5), -(3)/(8) , 0 , (4)/(5) , 1,1 (2)/(3) , sqrt(8), 3.01, pi , 8.47}`
From the given set find :
(i) set of rational numbers
(ii) set of irrational numbers
(iii) set of integers
set of non - negative integers

Answer

To solve the problem step by step, we will analyze the given universal set and categorize the numbers into rational numbers, irrational numbers, integers, and non-negative integers.

### Given Universal Set:
\[ \text{Universal Set} = \{ -6, -5\frac{3}{4}, -\sqrt{4}, -\frac{3}{5}, -\frac{3}{8}, 0, \frac{4}{5}, 1, 1\frac{2}{3}, \sqrt{8}, 3.01, \pi, 8.47 \} \]

### Step 1: Find the Set of Rational Numbers
**Definition**: Rational numbers are numbers that can be expressed in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).

1. **Identify each number**:
- \( -6 \) is rational (can be written as \( -6/1 \)).
- \( -5\frac{3}{4} = -\frac{20}{4} - \frac{3}{4} = -\frac{23}{4} \) is rational.
- \( -\sqrt{4} = -2 \) is rational.
- \( -\frac{3}{5} \) is rational.
- \( -\frac{3}{8} \) is rational.
- \( 0 \) is rational (can be written as \( 0/1 \)).
- \( \frac{4}{5} \) is rational.
- \( 1 \) is rational (can be written as \( 1/1 \)).
- \( 1\frac{2}{3} = \frac{5}{3} \) is rational.
- \( \sqrt{8} = 2\sqrt{2} \) is not rational (we will check this in the next step).
- \( 3.01 \) is rational (can be written as \( 301/100 \)).
- \( \pi \) is not rational.
- \( 8.47 \) is rational (can be written as \( 847/100 \)).

**Rational Numbers Set**:
\[ \text{Rational Numbers} = \{ -6, -\frac{23}{4}, -2, -\frac{3}{5}, -\frac{3}{8}, 0, \frac{4}{5}, 1, \frac{5}{3}, 3.01, 8.47 \} \]

### Step 2: Find the Set of Irrational Numbers
**Definition**: Irrational numbers cannot be expressed in the form \( \frac{p}{q} \).

1. **Identify each number**:
- \( \sqrt{8} \) is irrational (approximately \( 2.828 \)).
- \( \pi \) is irrational.

**Irrational Numbers Set**:
\[ \text{Irrational Numbers} = \{ \sqrt{8}, \pi \} \]

### Step 3: Find the Set of Integers
**Definition**: Integers include all whole numbers (positive, negative, and zero).

1. **Identify each number**:
- \( -6 \) is an integer.
- \( -\sqrt{4} = -2 \) is an integer.
- \( 0 \) is an integer.
- \( 1 \) is an integer.

**Integers Set**:
\[ \text{Integers} = \{ -6, -2, 0, 1 \} \]

### Step 4: Find the Set of Non-Negative Integers
**Definition**: Non-negative integers include all whole numbers greater than or equal to zero.

1. **Identify each number**:
- \( 0 \) is a non-negative integer.
- \( 1 \) is a non-negative integer.

**Non-Negative Integers Set**:
\[ \text{Non-Negative Integers} = \{ 0, 1 \} \]

### Final Results:
1. **Set of Rational Numbers**: \( \{ -6, -\frac{23}{4}, -2, -\frac{3}{5}, -\frac{3}{8}, 0, \frac{4}{5}, 1, \frac{5}{3}, 3.01, 8.47 \} \)
2. **Set of Irrational Numbers**: \( \{ \sqrt{8}, \pi \} \)
3. **Set of Integers**: \( \{ -6, -2, 0, 1 \} \)
4. **Set of Non-Negative Integers**: \( \{ 0, 1 \} \)