Let `A = {3, pi , sqrt(2) , - 5,3 + sqrt(7),2//7}`. The subset of A containig all the elements from it which are irrational number is :

Options

A. `{sqrt(2), 3 + sqrt(7)}`
B. `{3,pi 2//7 , - 5,3 + sqrt(7)}`
C. `{3,2//7,-5}`
D. `{3,-5}`

Answer

To find the subset of set \( A \) that contains all the irrational numbers, we will analyze each element of the set \( A \).

**Step 1: Identify the elements of set \( A \)**
The set \( A \) is given as:
\[ A = \{ 3, \pi, \sqrt{2}, -5, 3 + \sqrt{7}, \frac{2}{7} \} \]

**Step 2: Determine which elements are irrational**
We need to check each element to see if it is rational or irrational:

1. **3**: This is a rational number.
2. **\(\pi\)**: This is an irrational number.
3. **\(\sqrt{2}\)**: This is an irrational number.
4. **-5**: This is a rational number.
5. **\(3 + \sqrt{7}\)**: Since \(\sqrt{7}\) is irrational, \(3 + \sqrt{7}\) is also irrational.
6. **\(\frac{2}{7}\)**: This is a rational number.

**Step 3: List the irrational numbers**
From the analysis, the irrational numbers in set \( A \) are:
- \(\pi\)
- \(\sqrt{2}\)
- \(3 + \sqrt{7}\)

**Step 4: Form the subset of irrational numbers**
Thus, the subset of \( A \) containing all the irrational numbers is:
\[ \{ \pi, \sqrt{2}, 3 + \sqrt{7} \} \]

**Final Answer**: The subset of \( A \) containing all the irrational numbers is:
\[ \{ \pi, \sqrt{2}, 3 + \sqrt{7} \} \]

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