30Oct

The Cardinality of Sets

Every set has to have a number of elements in it- for example, the set {1,2,3,4,5} has five elements in it. The set {0} has 1 element in it. The set {Bob, Sue, Mary} has 3 elements in it. The number of elements in a set is called the Cardinal Number, and it is normally reffered to as cardinality. Every set has to have a cardinal number, be it 0, 1, 2, seven trillion, or even infinity.

Another way to state cardinality

Another common way to express “The cardinality of set X” is as nX, (n represents “The number of elements in”) where X is the name of the set. You may also substitute the actual set in, for examplen{1,2,3,4,5}=5, since there are five elements in a set.

A special instance of Cardinality

One of the ways to use cardinality besides simply counting the number of elements in a set is to find the number of elements in the union of a set without actually finding the union of the sets. To do this, you simply use the following formula, where A and B are the two sets who’s union you are trying to find the cardinality of (explained below):

n(A)+n(B)-n(A &cap B)

The formula may seem complicated, but it really isn’t. Remember that a union includes all the elements from each set. Therefore, you must count all the elements from set A, and all the elements from set B, but none of them may be counted twice. Therefore, you take the number of elemts in set A (the first part), add them to all the elements in set B (The second part) and then remove the elements that are in both (the third part). If this doesn’t make sense to you, don’t worry, you can always just count the number of elements in a union. It is, however, a good idea to know the formula in case it is on a test.

Examples

Cardinality of {January, February, March, April…. November, December} = 12
n{Monday, Tuesday, Wensday…. Saturday, Sunday} = 7
n{1,2,3,4,5…} = Infinity
The union of {January, February, March… November, December} and {Monday, Tuesday, … Saturday, Sunday} = (12)+(7)-(0)=21.

What to remember

  • Cardinality=The number of elements in a set
  • Can be expressed as n{Set Goes Here}
  • Know the formula for union of set cardinality

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