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The uncovered set can be thought of as requiring its candidates to have a two-step beatpath to every candidate not in the uncovered set. The Smith set requires a one-step beatpath (i.e. of at most two candidates, a direct pairwise victory).
 
'''[[Independence of covered alternatives''']] says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the [[uncovered set]]. Independence of covered alternatives implies [[Independence of Smith-dominated Alternatives]] (since independence of covered alternatives implies that one can eliminate everyone outside of the uncovered set without changing the winner, and the uncovered set is a subset of the Smith set, therefore eliminating everyone outside of the Smith set also can't change the winner), which further implies [[Smith criterion|Smith]] and thus [[Condorcet criterion|Condorcet]]. If a method is independent of covered alternatives, then the method fails monotonicity if perfect ties can always be broken in favor of a choice W by usingadding ballots ranking W first.
 
The uncovered set implies [[Pareto]], because Pareto implies that the Pareto-dominant candidate pairwise beats any candidates the Pareto-inferior candidate beats. This is because all voters rank the Pareto candidate equal to or better than the Pareto-inferior candidate. <ref>{{Cite web|url=https://www.researchgate.net/publication/225729286_Alternate_Definitions_of_the_Uncovered_Set_and_Their_Implications|title=Alternate Definitions of the Uncovered Set and Their Implications|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref>
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