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Number - Types of Number and Set Notation

Grade 9IGCSE

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

🔑Concepts

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Natural Numbers (ℕ): Counting numbers starting from 1 (1, 2, 3, ...).

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Integers (ℤ): Whole numbers, including negative, positive, and zero (..., -2, -1, 0, 1, 2, ...).

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Rational Numbers (ℚ): Numbers that can be written as a fraction a/b where a and b are integers and b ≠ 0. Includes terminating and recurring decimals.

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Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π). Their decimals are non-terminating and non-recurring.

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Real Numbers (ℝ): The set of all rational and irrational numbers.

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Prime Numbers: Numbers greater than 1 with exactly two factors (1 and itself). Note: 1 is not a prime number.

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Set Notation Symbols: ∈ (element of), ∉ (not an element of), ⊂ (subset), ∪ (union/OR), ∩ (intersection/AND), ξ (universal set), ∅ (empty set), A' (complement of A).

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

📐Formulae

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

N⊂Z⊂Q⊂R\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

x∈A  ⟹  x is a member of set Ax \in A \implies x \text{ is a member of set } A

A′={x:x∈ξ,x∉A}A' = \{x : x \in \xi, x \notin A\}

💡Examples

Problem 1:

Given the set S={−3,0,2,34,7,π,0.3˙}S = \{-3, 0, \sqrt{2}, \frac{3}{4}, 7, \pi, 0.\dot{3}\}, list the elements that are Irrational Numbers.

Solution:

{2,π}\{\sqrt{2}, \pi\}

Explanation:

2\sqrt{2} and π\pi cannot be expressed as fractions of two integers. −3,0,7-3, 0, 7 are integers (rational), 3/43/4 is a fraction (rational), and 0.3˙0.\dot{3} is a recurring decimal which equals 1/31/3 (rational).

Problem 2:

If ξ={x:x∈Z,1≤x≤10}\xi = \{x : x \in \mathbb{Z}, 1 \le x \le 10\}, A={prime numbers}A = \{\text{prime numbers}\} and B={even numbers}B = \{\text{even numbers}\}, find n(A∩B)n(A \cap B).

Solution:

1

Explanation:

First, define the sets: ξ={1,2,3,4,5,6,7,8,9,10}\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}. A={2,3,5,7}A = \{2, 3, 5, 7\}. B={2,4,6,8,10}B = \{2, 4, 6, 8, 10\}. The intersection A∩BA \cap B is the set of elements common to both, which is {2}\{2\}. The cardinality n(A∩B)n(A \cap B) is the count of elements, which is 1.

Problem 3:

Shade the region representing (A∪B)′∩C(A \cup B)' \cap C on a Venn Diagram.

Solution:

The region inside CC that does not overlap with AA or BB.

Explanation:

(A∪B)′(A \cup B)' represents everything outside the circles of AA and BB. The intersection ∩C\cap C restricts this area only to what is also inside circle CC.