Pot/Lid/Predicate logic/Exercise

We consider the sentences "For every pot there is a lid“ and "There is a lid for every pot“, which, in everyday life, are considered to mean the same. But when we understand the statements in the strict sense of predicate logic, reading them from the beginning to the end, two distinct meanings occur.

1. Formulate the statements with the help of additional words so that the two distinct meanings become apparent.
2. Let ${}T$ denote the set of pots and let ${}D$ be the set of lids. Let ${}P$ denote the binary predicate such that (for ${}x\in T$ and ${}y\in D$) ${}P(x,y)$ means that ${}y$ is suitable for ${}x$. Formulate both sentences with appropriate mathematical symbols.
3. Can we infer logically from the statement that for every pot there exists a lid, the statement that for every lid there exists a pot?
4. How can you explain that in everyday understanding, the two statements are considered to mean the same thing?

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