I'm trying to solve this question.
Let A be a set and P(x) and Q(x) be statements with one variable Are the statements
$$((∃x ∈ A)(P(x)))∨((∃x ∈ A)(Q(x)))$$ and$$(∃x ∈ A)(P(x)∨Q(x))$$
equivalent? If yes give a proof. Give a counterexample if false. But I'm currently stuck in how to prove they are equivalent, I don't know where to start. I'd appreciate the help.
In order to prove equivalence, you have show that statement 1 implies statement 2, and statement 2 implies statement 1. To begin, start with the assumption that $((\exists x \in A)(P(x)))\lor ((\exists x \in A)(Q(x)))$ and show that this implies $(\exists x \in A)(P(x)\lor Q(x))$. After, you assume the second and show that it implies the first.
An example of the start of the proof would be along the lines "Assume $((\exists x \in A)(P(x)))\lor ((\exists x \in A)(Q(x)))$, then..."