Difference between revisions of "Cardinality"

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(don't mix finite and infinite sets)
(start with finite sets - leading up to definition that applies for infinite sets)
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Cardinality is a measure of the size of a [[set theory|set]].  For [[finite]] sets, its cardinality is simply the number of elements in it.  
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Cardinality is a measure of the size of a [[set theory|set]].  For [[finite]] sets, its cardinality is simply the number of elements in it.
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For example, there are 7 days in the week (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday), so the cardinality of the set of days of the week is 7. Similarly, there are 26 letters in the [[alphabet]], so the set of letters of the alphabet has cardinality 26. As another example, the set of  suits for [[playing card]]s is 4 (there are four suits, hearts, diamonds, clubs and spades).
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One interesting property of finite sets is that when two sets have the same cardinality, they can be put into a one-to-one correspondence (known as a [[bijection]]). For example, there are 4 suits in a deck of cards, and four main compass directions (North, South, East and West). Thus, the cardinality of the set of suits in a deck of cards, and that of the set of compass directions, are both four, so the two sets have the same cardinality.
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We can place them in one-to-one correspondence as follows:
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Hearts  <-> North,
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Diamonds <-> South,
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Clubs <-> East,
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Spades <-> West. Each suit corresponds to exactly one compass direction, and vice versa.
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Similarly, we can put the set of days of the week, and the [[Seven Deadly Sins]] in one-to-one correspondence:
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Monday <-> Pride, Tuesday <-> Envy, Wednesday <-> Gluttony, Thursday <-> Lust, Friday <-> Anger, Saturday <-> Avarice, Sunday <-> Sloth.
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Also, if two finite sets have different cardinalities, they cannot be placed in a one-to-one correspondence. If we tried to place the 4 suits, and the 7 days of the week in such a correspondence, we could start with Hearts  <-> Monday, Diamonds <-> Tuesday, Clubs <-> Wednesday, Spades <-> Thursday. Then we would have to start repeating suits, which is not allowed for a one-to-one correspondence.
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This suggests another definition for cardinality, that two sets have the same cardinality if there is a one-to-one correspondence between them. This is the definition we will use for infinite sets, where the  concept of the number of elements of a set is not as clear.
  
 
[[category: set theory]]
 
[[category: set theory]]

Revision as of 14:53, August 19, 2007

Cardinality is a measure of the size of a set. For finite sets, its cardinality is simply the number of elements in it.

For example, there are 7 days in the week (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday), so the cardinality of the set of days of the week is 7. Similarly, there are 26 letters in the alphabet, so the set of letters of the alphabet has cardinality 26. As another example, the set of suits for playing cards is 4 (there are four suits, hearts, diamonds, clubs and spades).

One interesting property of finite sets is that when two sets have the same cardinality, they can be put into a one-to-one correspondence (known as a bijection). For example, there are 4 suits in a deck of cards, and four main compass directions (North, South, East and West). Thus, the cardinality of the set of suits in a deck of cards, and that of the set of compass directions, are both four, so the two sets have the same cardinality.

We can place them in one-to-one correspondence as follows: Hearts <-> North, Diamonds <-> South, Clubs <-> East, Spades <-> West. Each suit corresponds to exactly one compass direction, and vice versa.

Similarly, we can put the set of days of the week, and the Seven Deadly Sins in one-to-one correspondence: Monday <-> Pride, Tuesday <-> Envy, Wednesday <-> Gluttony, Thursday <-> Lust, Friday <-> Anger, Saturday <-> Avarice, Sunday <-> Sloth.

Also, if two finite sets have different cardinalities, they cannot be placed in a one-to-one correspondence. If we tried to place the 4 suits, and the 7 days of the week in such a correspondence, we could start with Hearts <-> Monday, Diamonds <-> Tuesday, Clubs <-> Wednesday, Spades <-> Thursday. Then we would have to start repeating suits, which is not allowed for a one-to-one correspondence.

This suggests another definition for cardinality, that two sets have the same cardinality if there is a one-to-one correspondence between them. This is the definition we will use for infinite sets, where the concept of the number of elements of a set is not as clear.