self doubt

Question

What is the English of both statement

∀x((x!=0)→∃y(xy=100))

∀x∃y((x!=0)→(xy=100))

Answer 1

Both the statements are equivalent. It is Null-quantification rule.

i.e. $\exists x(A \rightarrow P(x)) \equiv A \rightarrow \exists xP(x)$  ..(Where $A$ does not have $x$ as free variable)

So, the interpretation of both the statements is (assuming the domain is set of all real numbers) :

∀x((x!=0)→∃y(xy=100)) = For every $x$, if $x$ is non-zero then there exists some $y$ such that $xy$=100.

 

Answer 2

1. For every x, IF x is non-zero THEN there exists atleast 1 y such that product of x and y=100.
2. For every x,there exists atleast 1 y such that IF x is non-zero THEN product of x and y=100.

Note the placement of IF and THEN.  The "non-zero" condition on x is evaluated before declaring the existence of y in part 1.