Equivalent Statements of A ⊂ B
Question:
For any two sets \( A \) and \( B \), show that the following statements are equivalent:
\[ (i)\ A\subset B \] \[ (ii)\ A-B=\phi \] \[ (iii)\ A\cup B=B \] \[ (iv)\ A\cap B=A \]Solution
We prove the equivalence step by step.
(i) ⇒ (ii)
Assume:
\[ A\subset B \]Then every element of \( A \) belongs to \( B \).
Therefore, there is no element in \( A \) which is not in \( B \).
\[ A-B=\phi \](ii) ⇒ (iii)
Assume:
\[ A-B=\phi \]This means every element of \( A \) already belongs to \( B \).
Hence adding elements of \( A \) to \( B \) does not change \( B \).
\[ A\cup B=B \](iii) ⇒ (iv)
Assume:
\[ A\cup B=B \]This means every element of \( A \) is contained in \( B \).
Therefore, common elements of \( A \) and \( B \) are exactly the elements of \( A \).
\[ A\cap B=A \](iv) ⇒ (i)
Assume:
\[ A\cap B=A \]Every element of \( A \cap B \) belongs to both \( A \) and \( B \).
Since \( A\cap B=A \), every element of \( A \) belongs to \( B \).
Therefore,
\[ A\subset B \]Hence all the statements are equivalent.