I would like to know an efficient way of disproving existential quantifier ∃ to show that "for every value of a P(a) is false." ? Also, proving universal quantifier ∀ to show that "for every a, P(a) is true" seems difficult.
The question I'm looking at says the universe is all integers, where p(x, y) denotes "x evenly divides y." An example where the answer is true is: ∀y p(1, y)
We know that 1 divided by any number is 1, but how would I actually prove this? Another true statement is ∀y∃x p(x, y)
Best Answer
Let y be an arbitrary integer. Since 1 * y = y, we see that 1 divides y, so p(1,y) holds. Since y was arbitrary, we are done.
Not very exciting, admittedly, but basic idea is that we prove a universally quantified formula by proving it for an arbitrary element of the domain.