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Sets - Cardinal Number of a Set

Grade 7ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Cardinal Number of a set, denoted as n(A)n(A), is the total number of distinct elements present in a finite set AA. For example, if A={2,4,6,8}A = \{2, 4, 6, 8\}, then n(A)=4n(A) = 4. Visually, this is represented by counting each individual item inside the boundary of the set.

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A Finite Set has a countable number of elements, meaning its cardinal number is a whole number. In contrast, an Infinite Set has no fixed cardinal number because its elements continue indefinitely, often represented visually by an ellipsis (......) like the set of natural numbers N={1,2,3,...}N = \{1, 2, 3, ...\}.

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The Cardinal Number of an Empty Set (or Null Set), denoted by ∅\emptyset or {}\{\}, is always 00. This is written as n(∅)=0n(\emptyset) = 0. Visually, an empty set is represented by a circle or pair of brackets containing no symbols at all.

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A Singleton Set is a set that contains exactly one element. Its cardinal number is always 11. For example, if S={0}S = \{0\}, then n(S)=1n(S) = 1. Note that {0}\{0\} is not an empty set because it contains the number zero.

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For Disjoint Sets, which are sets that have no common elements, the cardinal number of their union is simply the sum of their individual cardinal numbers. Visually, disjoint sets are represented in a Venn diagram as two separate circles that do not touch or overlap.

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For Overlapping Sets, the cardinal number of the union n(A∪B)n(A \cup B) is calculated by adding the sizes of both sets and then subtracting the size of their intersection n(A∩B)n(A \cap B). Visually, this intersection is the shared football-shaped region where two circles overlap in a Venn diagram; we subtract it once because it is counted in both circle AA and circle BB.

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The cardinality of the difference of two sets, n(A−B)n(A - B), represents the number of elements that belong to set AA but NOT to set BB. Visually, this corresponds to the crescent-moon shape of circle AA after the overlapping part with circle BB has been removed.

📐Formulae

n(A)=number of elements in set An(A) = \text{number of elements in set } A

n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)

n(A∪B)=n(A)+n(B) (if A and B are disjoint sets)n(A \cup B) = n(A) + n(B) \text{ (if } A \text{ and } B \text{ are disjoint sets)}

n(A∩B)=n(A)+n(B)−n(A∪B)n(A \cap B) = n(A) + n(B) - n(A \cup B)

n(A−B)=n(A)−n(A∩B)n(A - B) = n(A) - n(A \cap B)

n(B−A)=n(B)−n(A∩B)n(B - A) = n(B) - n(A \cap B)

n(A)=n(A−B)+n(A∩B)n(A) = n(A - B) + n(A \cap B)

💡Examples

Problem 1:

Given the set P={x:x is a prime number less than 20}P = \{x : x \text{ is a prime number less than } 20\}, find the cardinal number n(P)n(P).

Solution:

Step 1: List the elements of the set PP in roster form. The prime numbers less than 2020 are 2,3,5,7,11,13,17,2, 3, 5, 7, 11, 13, 17, and 1919. Step 2: Write the set as P={2,3,5,7,11,13,17,19}P = \{2, 3, 5, 7, 11, 13, 17, 19\}. Step 3: Count the number of distinct elements in the set. Step 4: n(P)=8n(P) = 8.

Explanation:

To find the cardinal number, we first convert the set from set-builder notation to roster form and then count the elements.

Problem 2:

If n(A)=25n(A) = 25, n(B)=18n(B) = 18, and n(A∪B)=35n(A \cup B) = 35, find the number of elements in the intersection of AA and BB, i.e., n(A∩B)n(A \cap B).

Solution:

Step 1: Use the fundamental formula for the cardinality of the union of two sets: n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B) Step 2: Substitute the given values into the formula: 35=25+18−n(A∩B)35 = 25 + 18 - n(A \cap B) Step 3: Simplify the right side: 35=43−n(A∩B)35 = 43 - n(A \cap B) Step 4: Rearrange to solve for n(A∩B)n(A \cap B): n(A∩B)=43−35n(A \cap B) = 43 - 35 Step 5: Calculate the final result: n(A∩B)=8n(A \cap B) = 8

Explanation:

This problem uses the relationship between the union and intersection of two sets. By knowing three of the values, we can solve for the fourth using algebraic manipulation.