FOR ANY TWO SETS `Aa n dB` , show that the following statements are equivalent: `AsubB` (ii) `A-B=varphi` (iii) `AuuB=B` (iv) `AnnB=Adot`

Explanation

To show that the statements (i) \( A \subseteq B \), (ii) \( A - B = \varnothing \), (iii) \( A \cup B = B \), and (iv) \( A \cap B = A \) are equivalent, we will prove that each statement implies the next in a circular manner.

### Step 1: Prove \( A \subseteq B \) implies \( A - B = \varnothing \)

- **Proof**:
If \( A \subseteq B \), then every element \( x \) in \( A \) is also in \( B \). The set difference \( A - B \) consists of elements that are in \( A \) but not in \( B \). Since there are no elements in \( A \) that are not in \( B \), we conclude that \( A - B = \varnothing \).

### Step 2: Prove \( A - B = \varnothing \) implies \( A \subseteq B \)
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