Discrete Mathematics for Computer Science

158 CHAPTER 3 Relations

Table 3.1

SpecialDeck of

Cards

SpecialDeck

10 of Clubs

Jack of Clubs

King of Clubs

10 of Hearts

Jack of Hearts

Queen of Hearts

Much of the information about suits and card values is irrelevant to many card games.

Often, only two properties are important: whether two cards are in the same suit, and

whether one card is higher than another. Everything else, such as the names of the suits,

the names of the cards, and perhaps even the number of cards per suit, is unimportant. For

example, a person would immediately be able to translate the rules of many games to a

deck with five suits called red, yellow, blue, green, and black, with each suit consisting of

16 cards numbered 1, 2, 3,..., 16.

How can one abstract these important properties? Notice that both properties involve

a comparison between two objects. For example, given two cards a and b, does a have

a higher value than b? As a matter of convention, two elements that are related in some

special way are often represented by an ordered pair. If a is higher than b, then this could

be represented by the ordered pair (a, b). Formally, the relation HigherValue defined on

SpecialDeck is the set of ordered pairs (a, b) where the value of card a is higher than

the value of card b. This relation for SpecialDeck is shown in Table 3.2. Each ordered pair

(a, b) in the relation contributes one row to the table, with the higher-valued card appearing

first in the row and the lower-valued card second.

Table 3.2 HigherValue Relation

HigherValue

Jack of Hearts 10 of Hearts

Queen of Hearts 10 of Hearts

Jack of Clubs 10 of Hearts

King of Clubs 10 of Hearts

Queen of Hearts Jack of Hearts

King of Clubs Jack of Hearts

King of Clubs Queen of Hearts

Jack of Hearts 10 of Clubs

Queen of Hearts 10 of Clubs

Jack of Clubs 10 of Clubs

King of Clubs 10 of Clubs

Queen of Hearts Jack of Clubs

King of Clubs Jack of Clubs