krit.club logo

Number - Set Theory and Venn Diagrams

Grade 10IGCSE

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

🔑Concepts

•

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

•

Notation: ∈\in (is an element of), ∉\notin (is not an element of), ∅\emptyset or { } (empty set).

•

Universal Set (ξ\xi or E\mathcal{E}): The set containing all possible elements under consideration.

•

Union (A∪BA \cup B): Elements in set A, or set B, or both.

•

Intersection (A∩BA \cap B): Elements that are in both set A and set B.

•

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

•

Subsets: A⊆BA \subseteq B (A is a subset of B) and A⊂BA \subset B (A is a proper subset of B).

•

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

•

Number Sets: N\mathbb{N} (Natural numbers), Z\mathbb{Z} (Integers), Q\mathbb{Q} (Rational numbers), R\mathbb{R} (Real numbers).

📐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)

For disjoint sets (no overlap): n(A∪B)=n(A)+n(B)n(A \cup B) = n(A) + n(B)

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 \leq x \leq 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\}. Find n(A∩B)′n(A \cap B)'.

Solution:

7

Explanation:

First, find A∩BA \cap B, which are elements common to both: {2,4}\{2, 4\}. The complement (A∩B)′(A \cap B)' consists of all elements in ξ\xi except 2 and 4, which are {1,3,5,6,7,8,9,10}\{1, 3, 5, 6, 7, 8, 9, 10\}. Counting these elements, n(A∩B)′=8n(A \cap B)' = 8.

Problem 2:

In a class of 30 students, 18 study Spanish, 15 study French, and 5 study neither. How many students study both Spanish and French?

Solution:

8

Explanation:

Let SS be Spanish and FF be French. n(ξ)=30n(\xi) = 30. Students studying at least one language n(S∪F)=30−5=25n(S \cup F) = 30 - 5 = 25. Using the formula n(S∪F)=n(S)+n(F)−n(S∩F)n(S \cup F) = n(S) + n(F) - n(S \cap F), we get 25=18+15−n(S∩F)25 = 18 + 15 - n(S \cap F). Solving for n(S∩F)n(S \cap F): 25=33−n(S∩F)⇒n(S∩F)=33−25=825 = 33 - n(S \cap F) \Rightarrow n(S \cap F) = 33 - 25 = 8.

Problem 3:

Describe the shaded region in a Venn diagram where the area inside Circle A and Circle B is colored, but the overlapping middle section is left white.

Solution:

(A∪B)∩(A∩B)′(A \cup B) \cap (A \cap B)' or (A∩B′)∪(A′∩B)(A \cap B') \cup (A' \cap B)

Explanation:

The region represents elements in A or B but not in both. This is known as the symmetric difference. It can be written as the union minus the intersection, or the union of 'A only' and 'B only'.