Question

in the game of hearts, four players are each dealt 13 cards from a deck of 52. is this a function? if so, what sets make up the domain and codomain, and is the function injective, surjective, bijective, or neither?

Answer 1

Dealing 13 cards each to four players in the game of Hearts can be seen as an injective function with the domain being the set of players and the codomain the set of all possible 13-card combinations from the deck. The function is not surjective, as not all card combinations are represented in a single deal, and therefore it is not bijective.

Step-by-step explanation:

In the game of Hearts, dealing four players 13 cards each from a standard 52-card deck can be considered a function if we define the act of dealing cards as a function that assigns a specific hand to each player. In this context, a function is a relation where every input in the domain is related to exactly one output in the codomain.

The domain of this function would be the set of four players, since that's who we are dealing the cards to. The codomain would be the set of all possible 13-card combinations from the 52-card deck. Because every player receives a unique set of 13 cards, and no card can be in more than one player's hand at the same time (since we are dealing without replacement), this function is injective (or one-to-one).

However, since not all possible 13-card combinations will be dealt out in a single game (as there are more than four possible combinations), the function is not surjective (or onto). Thus, the function is not bijective, because it is not both injective and surjective. The function is simply an injective function.