Prove that the Inverse Distributed on Union

Question

I was asked to prove that;

(R∪P)⁻= R⁻∪P⁻; R and P be a relations between such arbitrary sets

However, I've proved it in the following way;

(a,b)∈(R∪P) ↔ (a,b)∈R ∧ (a,b)∈P ↔ (b,a)∈R⁻ ∧ (b,a)∈P⁻ ↔ (b,a)∈(R⁻∪P⁻) ↔ (b,a)∈(R∪P)⁻

Since that (R∪P)⁻ and (R⁻∪P⁻) are equivalent, based on definition of equivalence between sets.

I have no idea if my preceding proof is correct or not!!

Answer 1

Your proof is not correct. In your attempt there are essentially two errors:

1. set union $\cup$ has the logical meaning of a "or", not of a "and";
2. your proof should start from an element in $(R \cup P)^T$ and not in $R \cup P$ (maybe this is just a typo).

A correct proof is the following: $$(b,a) \in (R \cup P)^T$$ is equivalent to $$(a,b) \in (R \cup P)$$ which is equivalent to $$(a,b)\in R \text{ or } (a,b) \in P$$ (and not to "$(a,b)\in R$ AND $(a,b) \in P$"), which is equivalent to $$(b,a)\in R^T \text{ or } (b,a) \in P^T$$ which is equivalent to $$(b,a) \in R^T \cup P^T$$