I. P and Q are reflexive \(\Rightarrow P \cap Q\) is reflexive.
II. P and Q are symmetric \(\Rightarrow P \cup Q\) is symmetric.
III. P and Q are transitive \(\Rightarrow P \cap Q\) is transitive.
Select the answer using the code given below.
This question explores the properties of binary relations, specifically focusing on how reflexive, symmetric, and transitive properties behave when relations are combined using intersection (\(\cap\)) and union (\(\cup\)). Let's analyze each statement:
Reflexive Properties Explained
A relation P on a set A is reflexive if for every element \(a \in A\), the pair \((a, a)\) is in P. That is, \(\forall a \in A, (a, a) \in P\). The statement asks if the intersection of two reflexive relations, \(P \cap Q\), is also reflexive.
- Given: P is reflexive, meaning \(\forall a \in A, (a, a) \in P\).
- Given: Q is reflexive, meaning \(\forall a \in A, (a, a) \in Q\).
- Consider the intersection \(P \cap Q\). For an element \((a, a)\) to be in \(P \cap Q\), it must be present in both P and Q.
- Since \((a, a) \in P\) and \((a, a) \in Q\) for all \(a \in A\), it follows that \((a, a) \in P \cap Q\) for all \(a \in A\).
- Conclusion: Therefore, \(P \cap Q\) is reflexive. Statement I is correct.
Symmetric Properties Explained
A relation P on a set A is symmetric if whenever \((a, b) \in P\), then \((b, a) \in P\). That is, \(\forall a, b \in A, (a, b) \in P \Rightarrow (b, a) \in P\). The statement asks if the union of two symmetric relations, \(P \cup Q\), is also symmetric.
- Given: P is symmetric, meaning \((a, b) \in P \Rightarrow (b, a) \in P\).
- Given: Q is symmetric, meaning \((a, b) \in Q \Rightarrow (b, a) \in Q\).
- Consider the union \(P \cup Q\). Let's assume \((a, b) \in P \cup Q\). This means either \((a, b) \in P\) or \((a, b) \in Q\) (or both).
- Case 1: If \((a, b) \in P\). Since P is symmetric, \((b, a) \in P\). Because \(P \subseteq P \cup Q\), we know that \((b, a) \in P \cup Q\).
- Case 2: If \((a, b) \in Q\). Since Q is symmetric, \((b, a) \in Q\). Because \(Q \subseteq P \cup Q\), we know that \((b, a) \in P \cup Q\).
- In both possible cases, if \((a, b) \in P \cup Q\), then \((b, a) \in P \cup Q\).
- Conclusion: Therefore, \(P \cup Q\) is symmetric. Statement II is correct.
Transitive Properties Explained
A relation P on a set A is transitive if whenever \((a, b) \in P\) and \((b, c) \in P\), then \((a, c) \in P\). That is, \(\forall a, b, c \in A, ((a, b) \in P \land (b, c) \in P) \Rightarrow (a, c) \in P\). The statement asks if the intersection of two transitive relations, \(P \cap Q\), is also transitive.
- Given: P is transitive, meaning \(((a, b) \in P \land (b, c) \in P) \Rightarrow (a, c) \in P\).
- Given: Q is transitive, meaning \(((a, b) \in Q \land (b, c) \in Q) \Rightarrow (a, c) \in Q\).
- Consider the intersection \(P \cap Q\). Let's assume \((a, b) \in P \cap Q\) and \((b, c) \in P \cap Q\).
- From \((a, b) \in P \cap Q\), we know \((a, b) \in P\) AND \((a, b) \in Q\).
- From \((b, c) \in P \cap Q\), we know \((b, c) \in P\) AND \((b, c) \in Q\).
- Now consider relation P: We have \((a, b) \in P\) and \((b, c) \in P\). Since P is transitive, it must be that \((a, c) \in P\).
- Next, consider relation Q: We have \((a, b) \in Q\) and \((b, c) \in Q\). Since Q is transitive, it must be that \((a, c) \in Q\).
- Since we've established that \((a, c) \in P\) and \((a, c) \in Q\), it follows directly that \((a, c) \in P \cap Q\).
- Conclusion: Therefore, \(P \cap Q\) is transitive. Statement III is correct.
Final Summary
All three statements are correct:
- Statement I: Intersection of reflexive relations is reflexive.
- Statement II: Union of symmetric relations is symmetric.
- Statement III: Intersection of transitive relations is transitive.
Thus, statements I, II, and III are all correct.
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