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Question
Given a non-empty set X, define the relation R in P(X) as follows:
For A, B ∈ P(X), (4, B) ∈ R iff A ⊂ B. Prove that R is reflexive, transitive and not symmetric.
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Solution
Let A ∈ P(X).
Then A ⊂ A
⇒ (A, A) ∈ R
Hence, R is reflexive.
Let A, B, C ∈ P(X) such that (A, B), (B, C) ∈ R
⇒ A ⊂ B, B ⊂ C
⇒ A ⊂ C
⇒ (A, C) ∈ R
Hence, R is transitive.
Φ, X ∈ P(X) such that Φ ⊂ X.
Hence (Φ, X) ∈ R. But, X ⊄ Φ, which implies that (X, Φ) ∉ R.
Thus, R is not symmetric.
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