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Number - Set notation and Venn diagrams

Grade 11IGCSE

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

🔑Concepts

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Definition of a Set: A collection of distinct objects called elements or members.

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Universal Set (ξ): The set containing all possible elements under consideration.

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Empty Set (∅ or {}): A set containing no elements.

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Intersection (A ∩ B): Elements that are in both Set A AND Set B.

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Union (A ∪ B): Elements that are in Set A OR Set B (or both).

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Complement (A'): Elements in the universal set that are NOT in Set A.

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Subset (A ⊆ B): Every element of A is also an element of B.

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Proper Subset (A ⊂ B): A is a subset of B, but A is not equal to B.

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Cardinality (n(A)): The number of elements in set A.

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Disjoint Sets: Sets with no elements in common (A ∩ B = ∅).

📐Formulae

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

n(A)+n(A′)=n(ξ)n(A) + n(A') = n(\xi)

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

Number of subsets of a set with nn elements = 2n2^n

💡Examples

Problem 1:

Given ξ={x:x is an integer, 1≤x≤10}\xi = \{x : x \text{ is an integer, } 1 \le x \le 10\}, A={2,4,6,8,10}A = \{2, 4, 6, 8, 10\}, and B={1,2,3,4,5}B = \{1, 2, 3, 4, 5\}. List the elements of (A∪B)′(A \cup B)'.

Solution:

(A∪B)′={7,9}(A \cup B)' = \{7, 9\}

Explanation:

First, find the union: A∪B={1,2,3,4,5,6,8,10}A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\}. The complement (A∪B)′(A \cup B)' consists of elements in the universal set ξ={1,2,3,4,5,6,7,8,9,10}\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} that are not in the union. The missing integers are 7 and 9.

Problem 2:

In a group of 40 students, 25 like Mangoes (M), 18 like Apples (A), and 5 like neither. Find the number of students who like both fruit.

Solution:

n(M∩A)=8n(M \cap A) = 8

Explanation:

Total students n(ξ)=40n(\xi) = 40. Students who like at least one fruit n(M∪A)=40−5=35n(M \cup A) = 40 - 5 = 35. Using the formula n(M∪A)=n(M)+n(A)−n(M∩A)n(M \cup A) = n(M) + n(A) - n(M \cap A), we get 35=25+18−n(M∩A)35 = 25 + 18 - n(M \cap A). Solving for the intersection: 35=43−n(M∩A)⇒n(M∩A)=43−35=835 = 43 - n(M \cap A) \Rightarrow n(M \cap A) = 43 - 35 = 8.

Problem 3:

Describe the shaded region in a Venn diagram where the area inside Circle A is shaded, but the overlap with Circle B is left white.

Solution:

A∩B′A \cap B'

Explanation:

The region belongs to A but specifically excludes any part of B. This is expressed as the intersection of A and the complement of B (B′B'). It can also be written as A∖BA \setminus B (A minus B).