Are these quantifier proofs correct? You just have to take counterexamples to disprove, and examples to prove right?
(i) Let . Take , so the statement is true.
(ii) Let . Take , so the statement is true.
(iii) For , so the statement is false.
(iv) For , so the statement is false.
(v) If .
If , take , so the statement is false.
(vi) Let . Take , so the statement is true.
(vii) Take , so the statement is true.
(viii) Take and the statement is true.
(ix) is false, so the statement is false.
(x) is true (see (i)).
is true (take ), so the statement is true.