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Sets: Playing Cards Daniel Hunnicutt Set Definition A set is an unordered collection of objects usually denoted by S = { } Playing Card Set (P): {4 of clubs, Ace of Hearts, …} Cardinality (number of elements): 52 Member element S = {a, b, c, …} Membership: a  P Subsets

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Sets: Playing Cards

Daniel Hunnicutt

set definition
Set Definition
  • A set is an unordered collection of objects usually denoted by S = { }
  • Playing Card Set (P):

{4 of clubs, Ace of Hearts, …}

Cardinality (number of elements): 52

Member element S = {a, b, c, …}

Membership: a  P

subsets
Subsets
  • A subset is any set that is contained within the original set

i.e. Subset Facecards = {King of clubs, …}

Facecards  P

Subset Diamonds ={2 of diamonds,…}

Diamonds  P

  • Power set: The set of all subsets of P

i.e. P(P) = { {}, {2 of Hearts}, …}

connectivity c
Connectivity , , -, c
  • Diamonds  Facecards:

{Jack of Diamonds, Queen of D, King of D, Ace of D}

  • Diamonds  Facecards:

Any Diamond or Facecard {King of clubs, 2 of diamonds,…}

  • Diamonds c: {all cards not Diamond}
  • Facecards – Diamonds:

{all facecards that are not diamonds}

partitions
Partitions
  • A partition of a set P is a set of nonempty subsets of P such that every element c in P is in exactly one of these subsets.
  • Example: Partition of Suits

{Diamonds, Clubs, Hearts, Spades}

further application card games
Further Application:Card Games
  • Poker:Object of poker is to obtain the highest valued subset.

Subsets include: 4 of a kind, 3 of a kind, Straight, Full House, Flush, Royal Flush, etc.}

  • Hearts: Game is initially partitioned evenly (13 cards each), avoid certain subsets ({Hearts}, {Queen of Spades})
further application cont
Further Application: cont.
  • Solitaire: Collect an ordered subset, beginning with Ace, partitioned into stacks
  • Almost all card games contain some elements of sets and set theory.