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The task is to verify the truth of the given statements about binary relations $P, P_1, P_2$:

1. $\overline{P_1 \cup P_2} = \overline{P_1} \cap \overline{P_2}$ (De Morgan's law for sets).
2. $\overline{P_1 \cap P_2} = \overline{P_1} \cup \overline{P_2}$ (De Morgan's law for sets).
3. $P_1^{-1} \cup P_2^{-1} = (P_1 \cup P_2)^{-1}$ (inverse of the union).

Let me verify each of these statements mathematically.

Verification:

(a) $\overline{P_1 \cup P_2} = \overline{P_1} \cap \overline{P_2}$

This is a standard application of De Morgan's law in set theory, which states that: $\overline{A \cup B} = \overline{A} \cap \overline{B}$ Thus, statement $(a)$ is true.

(b) $\overline{P_1 \cap P_2} = \overline{P_1} \cup \overline{P_2}$

This is another standard application of De Morgan's law, which states: $\overline{A \cap B} = \overline{A} \cup \overline{B}$ Thus, statement $(b)$ is true.

(c) $P_1^{-1} \cup P_2^{-1} = (P_1 \cup P_2)^{-1}$

To verify this, consider the inverse of a relation:

• If $P^{-1} = \{ (y, x) : (x, y) \in P \}$, then: $(P_1 \cup P_2)^{-1} = P_1^{-1} \cup P_2^{-1}$ This holds true because the inverse operation distributes over unions.

Thus, statement $(c)$ is true.

Conclusion:

All three statements $(a), (b), (c)$ are true based on set theory and binary relation properties.

If you have further questions or need a detailed breakdown of any specific part, feel free to ask!

Related Questions:

1. What is the formal definition of the inverse of a binary relation?
2. How do De Morgan's laws apply to sets and relations?
3. Can you provide examples of binary relations for which these properties hold?
4. How does the complement operation interact with set unions and intersections?
5. What are practical applications of these properties in computer science or mathematics?

Tip: When working with binary relations, always visualize sets as pairs or draw diagrams to better understand operations like union, intersection, and complement.