Cardinality of SetThe 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 \( A = \{x: x< 4, x \in \mathbb{W} \}\) 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.
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,
- The cardinality of finite set A is \(n(A)\)
- The cardinality of countable set is \(\aleph_0\) (read as aleph-naught or aleph-zero or aleph-null)
- The cardinality of uncountable set is \(𝑐\) (read as continuum)
Set Operation and CardinalityStudy the given Venn-diagram, and find the elements and cardinality of tabulated sets.
| SN | Set Notation | |||
| 1 | \(A_o\) | \(A_o=\{a,b\}\) or \(A-B=\{a,b\}\) \(n_o(A)=2\) or \(n(A-B)=2\) | \(A_o=\{1\}\) or \(A-B=\{1\}\) \(n_o(A)=1\) or \(n(A-B)=1\) | \(A_o=\{1,2,3\}\) or \(A-B=\{1,2,3\}\) \(n_o(A)=3\) or \(n(A-B)=3\) |
| 2 | \(B_o\) | |||
| 3 | \(A \cap B\) | |||
| 4 | \(\overline{A \cup B}\) | |||
| 5 | \(A\) | |||
| 6 | \(\overline{A}\) | |||
| 7 | \(B\) | |||
| 8 | \(\overline{B}\) | |||
| 9 | \(A \triangle B\) | |||
| 10 | \(\overline{A \triangle B}\) | |||
| 11 | \(A \cup B\) | |||
| 12 | \(\overline{A _o}\) | |||
| 13 | \(\overline{B_o}\) | |||
| 14 | \(\overline{A \cap B}\) |
Arithmetics on 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:
-
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) -
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) - 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)
| SN | Set Notation | |||
| 1 | \(n_o(A)\) who like only A |
\(n_o(A)=p\) |
\(n_o(A)=w\) |
\(n_o(A)=a\) |
| 2 | \(n_o(B)\) who like only B |
\(n_o(B)=q\) |
\(n_o(B)=0\) |
\(n_o(B)=b\) |
| 3 | \(n(A \cap B)\) who like A and B both |
\(n(A \cap B)=r\) |
\(n(A \cap B)=x\) |
\(n(A \cap B)=0\) |
| 4 | \(n(\overline{A \cup B})\) who like Neither A nor B |
\(n(\overline{A \cup B})=s\) |
\(n(\overline{A \cup B})=y\) |
\(n(\overline{A \cup B})=c\) |
| 5 | \(n(A)\) who like A |
\(n(A)=p+r\) |
\(n(A)=x+w\) |
\(n(A)=a\) |
| 6 | \(n(\overline{A})\) who does not like A |
\(n(\overline{A})=q+s\) |
\(n(\overline{A})=y\) |
\(n(\overline{A})=b+c\) |
| 7 | \(n(B)\) who like B |
\(n(B)=q+r\) |
\(n(B)=x\) |
\(n(B)=b\) |
| 8 | \(n(\overline{B})\) who does not like B |
\(n(\overline{B})=p+s\) |
\(n(\overline{B})=w+y\) |
\(n(\overline{B})=a+c\) |
| 9 | \(n(A \triangle B)\) who like exactly one |
\(n(A \triangle B)=p+q\) |
\(n(A \triangle B)=w\) |
\(n(A \triangle B=a+b\) |
| 10 | \(n(\overline{A \triangle B})\) |
\(n(\overline{A \triangle B})=r+s\) |
\(n(\overline{A \triangle B})=x+y\) |
\(n(\overline{A \triangle B})=c\) |
| 11 | \(n(A \cup B)\) who like either A or B who like at least one |
\(n(A \cup B)=p+q+r\) |
\(n(A \cup B)=x+w\) |
\(n(A \cup B)=a+b\) |
| 12 | \(n(\overline{A _o})\) Except who like A only |
\(n(\overline{A _o})=q+r+s\) |
\(n(\overline{A_o})=x+y\) |
\(n(\overline{A_o})=b+c\) |
| 13 | \(n(\overline{B_o})\) Except who like B only |
\(n(\overline{B_o})=p+r+s\) |
\(n(\overline{B_o})=w+x+y\) |
\(n(\overline{B_o})=a+c\) |
| 14 | \(n(\overline{A \cap B})\) Except who like both who like at most one |
\(n(\overline{A \cap B})=p+q+s\) |
\(n(\overline{A \cap B})=w+y\) |
\(n(\overline{A \cap B})=a+b+c\) |
Test your understandings
Test your Understandings
From the Venn-diagram given below, find the cardinal number given sets.
- \(n(\phi) \)
- \(n(A-B) \)
- \(n(B-A) \)
- \(n(A \cap B) \)
- \(n(A \cup B)' \)
- \(n(A) \)
- \(n(B) \)
- \(n(A') \)
- \(n(B)' \)
- \(n(A \triangle B) \)
- \(n(A \triangle B)' \)
- \(n(AUB) \)
- \(n(A_0)' \)
- \(n(B_0)' \)
- \(n(A \cap B)' \)
- \(n(U) \)
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