So the proposition is true in all three domains.
(i) and (ii) are correct.
i) false x = 3
ii) false x = 3
iii) false x = 2 ?
But (iii) is a bit sneaky, because in the domain is false for all . So for every value of that makes true, is true. In other words, there isn't a counterexample, so it's a true proposition.
These are all correct.
No. If an existential statement is false, it simply means that no value of exists that makes the statement true.Is it possible to find counterexamples for the existential statements?
No. When you form a proposition like you are simply saying if is true, then is true. You aren't saying anything at all about actually being true at all.Also, in the first two problems with the universal quantifier, for these to be true, would p(x) and q(x) have to hold for every value in the domain?
Indeed may never be true for any values of - see for example part (iii) of the statement , where is false for all values of in the domain .