There are two good ways to prove this:
The first is an equational sequence.
The second uses two deductive sequences:
We first transform the conclusion by pushing the ¬ down as far as possible. We do this so that the conclusion will begin with a quantifier, and we then know which quantifier rule to try. The transformed version is:
∀w:∃x:[(x∈L)∧(w≠xy)]
Is this conclusion true without the first assumption? (The second assumption is true for strings over any nonempty alphabet.)
No, it is not necessarily true. Without the first assumption, we don't know anything about L. If L were the empty set, for example, we could never find an x such that x∈L, and so we could never make the conclusion true. But for some other L it could still work.
Last modified 22 February 2006