In the following sets of reactants which two sets best exhibit the amphoteric character of Al$$_2$$O$$_3$$ · xH$$_2$$O?
Set I: Al$$_2$$O$$_3$$ · xH$$_2$$O$$_{(s)}$$ + OH$$^-_{(aq)}$$
Set II: Al$$_2$$O$$_3$$x · H$$_2$$O$$_{(s)}$$ + H$$_2$$O$$_{(l)}$$
Set III: Al$$_2$$O$$_3$$x · H$$_2$$O$$_{(s)}$$ + H$$^+_{(aq)}$$
Set IV: Al$$_2$$O$$_3$$ · xH$$_2$$O$$_{(s)}$$ + NH$$_{3(aq)}$$

Amphoteric substances can react with both acids and bases. Aluminum oxide hydrate, written as Al2O3·xH2O, is essentially aluminum hydroxide, Al(OH)3, which is amphoteric. To exhibit amphoteric character, it must react with an acid (like H+) to show basic behavior and with a base (like OH-) to show acidic behavior.

Let's analyze each set:

Set I: Al2O3·xH2O(s) + OH-(aq)
Here, OH- is a strong base. Aluminum hydroxide acts as an acid and reacts with OH- to form the aluminate ion. The reaction is: $$ \text{Al(OH)}_3 + \text{OH}^- \rightarrow [\text{Al(OH)}_4]^- $$ This demonstrates the acidic behavior (reaction with a base) of the amphoteric substance.

Set II: Al2O3·xH2O(s) + H2O(l)
Water is neutral. Aluminum hydroxide is insoluble and does not react significantly with water. This does not demonstrate amphoteric behavior because amphoterism requires reaction with an acid or a base, not a neutral substance.

Set III: Al2O3·xH2O(s) + H+(aq)
Here, H+ is an acid. Aluminum hydroxide acts as a base and reacts with H+ to form aluminum ions and water. The reaction is: $$ \text{Al(OH)}_3 + 3\text{H}^+ \rightarrow \text{Al}^{3+} + 3\text{H}_2\text{O} $$ This demonstrates the basic behavior (reaction with an acid) of the amphoteric substance.

Set IV: Al2O3·xH2O(s) + NH3(aq)
Ammonia (NH3) is a weak base. Aluminum hydroxide does not dissolve in ammonia because ammonia is not strong enough to form the soluble aluminate complex. The reaction only occurs with strong bases like OH-. Hence, no significant reaction occurs, and this set does not exhibit amphoteric behavior.

Sets I and III demonstrate the amphoteric character: Set I shows reaction with a base (acidic behavior), and Set III shows reaction with an acid (basic behavior). Sets II and IV do not show amphoteric behavior.

The options are:
A. Sets 2 and 4
B. Sets 1 and 2
C. Sets 1 and 3
D. Sets 3 and 4

Hence, the correct answer is Option C.