Cardinality of Set

Cardinality of Set

The concept and notation of Cardinality are due to Georg Cantor who defined the notion of cardinality and realized that sets can have different cardinalities. In summary,

  1. The cardinality of finite set A is \(n(A)\)
  2. The cardinality of countable set is \(\aleph_0\) (read as aleph-naught or aleph-zero or aleph-null)
  3. The cardinality of uncountable set is \(𝑐\) (read as continuum)

The cardinality of a set A is the number of elements of the set A . The cardinality of a set A is usually denoted by n(A) but it can also be denoted as Card(A). For example:

  • If \( A = \{x: x< 4, x \in W \}\) then A = {0, 1, 2, 3} and n (A) = 4
  • If B = { letters in the word “mathematics”} then B = {m, a, t, h, e, i, c, s} and n(B) = 8.

Arithmetic of Cardinality

Arithmetic of cardinality in sets refers to the mathematical operations that involve counting the number of elements (cardinality) within sets. When solving verbal problems involving sets, you might encounter situations where you need to perform arithmetic operations such as addition, subtraction, multiplication, and division on the cardinalities of sets to find the desired information.
Here are some common scenarios where arithmetic of cardinality comes into play when solving verbal problems related to sets:

  1. Union of Sets: When we need to find the total number of elements in the union of two or more sets, we use the concept of cardinality. For example, if we have sets A and B, the cardinality of their union (A ∪ B) can be calculated by adding the cardinalities of A and B and then subtracting the cardinality of their intersection (A ∩ B) to avoid double counting any shared elements.
    n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
  2. Complements: The complement of a set A with respect to a larger set U (universal set) contains all elements in U that are not in A. You can calculate the cardinality of the complement by subtracting the cardinality of A from the cardinality of U.
    n(A') = n(U) - n(A)
  3. Subtraction of Sets: When you want to find the number of elements in one set that are not in another set, you can use subtraction of cardinalities. For instance, if you have sets A and B, the cardinality of the difference A - B is found by subtracting the cardinality of B from the cardinality of A.
    n(A - B) = n(A) - n(B)

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