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Question
Prove that:
\[A \subseteq B, B \subseteq C \text{ and } C \subseteq A \Rightarrow A = C .\]
Solution
\[Let x \in A\]
\[ \Rightarrow x \in B \left( \because A \subseteq B \right)\]
\[ \Rightarrow x \in C \left( \because B \subseteq C \right)\]
\[ \therefore x \in A \Rightarrow x \in C\]
\[ \Rightarrow A \subseteq C . . . \left( 1 \right)\]
\[\text{ It is given tha }, \]
\[C \subseteq A . . . (2)\]
\[\text{ From } \left( 1 \right) \text{ and } \left( 2 \right), \text{ we have }\]
\[A = C\]
\[\]
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