If `P={x:x in W " and " 4 le x le 8}` and `Q={x:x in N " and " x lt 6}`. Find :
(i) `P cup Q` and `P cap Q`
(ii) Is `(P cup Q) sup (P cap Q)` ?

Answer

To solve the given problem, we will follow these steps:

### Step 1: Define the sets P and Q

- **Set P**: This set consists of whole numbers \( x \) such that \( 4 \leq x \leq 8 \).

Therefore, \( P = \{4, 5, 6, 7, 8\} \).

- **Set Q**: This set consists of natural numbers \( x \) such that \( x < 6 \).

Therefore, \( Q = \{1, 2, 3, 4, 5\} \).

### Step 2: Find \( P \cup Q \) (Union of P and Q)

The union of two sets includes all the elements from both sets without repetition.

- \( P \cup Q = \{4, 5, 6, 7, 8\} \cup \{1, 2, 3, 4, 5\} \)

Combining these sets, we get:

- \( P \cup Q = \{1, 2, 3, 4, 5, 6, 7, 8\} \)

### Step 3: Find \( P \cap Q \) (Intersection of P and Q)

The intersection of two sets includes only the elements that are present in both sets.

- \( P \cap Q = \{4, 5, 6, 7, 8\} \cap \{1, 2, 3, 4, 5\} \)

The common elements are:

- \( P \cap Q = \{4, 5\} \)

### Step 4: Check if \( P \cup Q \) is a subset of \( P \cap Q \)

We need to determine if \( P \cup Q \subseteq P \cap Q \).

- We found \( P \cup Q = \{1, 2, 3, 4, 5, 6, 7, 8\} \)
- We found \( P \cap Q = \{4, 5\} \)

Since \( P \cup Q \) contains elements (1, 2, 3, 6, 7, 8) that are not in \( P \cap Q \), we conclude that:

- \( P \cup Q \) is **not** a subset of \( P \cap Q \).

### Final Answers:

1. \( P \cup Q = \{1, 2, 3, 4, 5, 6, 7, 8\} \)
2. \( P \cap Q = \{4, 5\} \)
3. No, \( P \cup Q \) is not a subset of \( P \cap Q \).

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