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Question
If A is any set, prove that: \[A \subseteq \phi \Leftrightarrow A = \phi .\]
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Solution
To prove: \[A \subseteq \phi \Leftrightarrow A = \phi\]
Proof:
Let: \[A \subseteq \phi\]
If A is a subset of an empty set, then A is the empty set.
∴\[A \subseteq \phi\]
Now, let
\[A = \phi\]
This means that A is an empty set.
We know that every set is a subset of itself.
\[A = \phi\]
Thus, we have \[A \subseteq \phi \Leftrightarrow A = \phi\]
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