A Level CS Chapter 21 Maths for Regular Expressions (Kalpesh)

What is a set in mathematics?

• An ordered list of values where repetition is allowed
• A collection of unordered, non-repeated data items of the same type
• A sequence of numbers arranged from smallest to largest
• A table of key-value pairs

Answer explanation

A SET is a collection of UNORDERED, NON-REPEATED data items. Key properties: (1) UNORDERED — {1,2,3} and {3,1,2} are the same set; (2) NO DUPLICATES — {1,1,2} is not a valid set, it would just be {1,2}; (3) Same type — all elements belong to the same category. Sets are fundamental to mathematics and computer science and appear in most programming languages (Python's set, Java's HashSet, etc.).

Which of the following correctly represents the set of natural numbers from 1 to 5?

• [1, 2, 3, 4, 5]
• (1, 2, 3, 4, 5)
• {1, 2, 3, 4, 5}
• <1, 2, 3, 4, 5>

Answer explanation

Sets are written using CURLY BRACES { }. So the set of natural numbers 1 to 5 is written {1, 2, 3, 4, 5}. Square brackets [ ] are used for lists/arrays (ordered, allows duplicates). Round brackets ( ) are used for tuples or coordinates. Angle brackets < > are used in BNF notation. Always use curly braces when writing sets in mathematics and computer science.

What are 'natural numbers' in the context of sets?

• All real numbers including decimals
• All positive whole numbers (1, 2, 3, 4, 5, ...)
• All integers including negatives
• All numbers between 0 and 1

Answer explanation

NATURAL NUMBERS (ℕ) are positive whole numbers: 1, 2, 3, 4, 5, ... The symbol ℕ represents the set of all natural numbers. Note: some definitions include 0, but in the AQA textbook context natural numbers start at 1. Natural numbers are important because the set ℕ is a COUNTABLY INFINITE set — you can list them one by one, even though the list never ends.

What does the symbol ∈ mean in set notation?

• Is not a member of
• Is a subset of
• Is a member of / belongs to
• Is equal to

Answer explanation

The symbol ∈ means 'IS A MEMBER OF' or 'belongs to'. For example: 3 ∈ {1, 2, 3, 4, 5} means '3 is a member of this set' — TRUE. 6 ∈ {1, 2, 3, 4, 5} means '6 is a member of this set' — FALSE. The symbol ∉ means 'is NOT a member of'. A 'member' or 'element' describes a value or item that belongs to a set.

If A = {1, 2, 3, 4, 5}, which of the following statements are TRUE?

• 3 ∈ A
• 6 ∈ A
• 1 ∈ A
• 0 ∈ A

Answer explanation

3 ∈ A is TRUE — 3 is in the set {1,2,3,4,5}. 1 ∈ A is TRUE — 1 is in the set. 6 ∈ A is FALSE — 6 is not in the set. 0 ∈ A is FALSE — 0 is not in the set. The ∈ symbol is essential AQA set notation — expect to see questions asking you to evaluate whether elements belong to a given set.

What is set comprehension?

• Reading and understanding a set definition
• The process of building a set using an expression or rule, rather than listing every element
• Combining two sets together
• Checking whether an element belongs to a set

Answer explanation

SET COMPREHENSION is defining a set using a RULE or EXPRESSION rather than listing every element. This is essential for large or infinite sets. The format is: A = {x | condition on x} which reads 'the set of all x such that [condition]'. For example: {x | x ∈ ℕ, x > 3} = {4, 5, 6, 7, ...}. The | symbol here means 'such that'. Most programming languages support this — Python calls it a 'set comprehension': {x for x in range(10) if x % 2 == 0}.

What does the set A = {x | x ∈ ℕ, x ≤ 5} represent?

• {0, 1, 2, 3, 4, 5}
• {1, 2, 3, 4, 5}
• {5, 6, 7, 8, ...}
• All natural numbers

Answer explanation

Reading the comprehension: 'the set of all x where x is a natural number AND x is less than or equal to 5'. Natural numbers start at 1, so: {1, 2, 3, 4, 5}. If it included 0 (if zero were a natural number in this context), it would be {0,1,2,3,4,5}. Note: ℕ (natural numbers) are positive whole numbers starting from 1 in AQA's definition. ℤ (integers) would include negative numbers and zero.

Which set comprehension defines all even natural numbers?

• {x | x ∈ ℕ, x = 2}
• {x | x ∈ ℕ, x mod 2 = 0}
• {x | x ∈ ℕ, x < 2}
• {x | x ∈ ℕ, x + 2}

Answer explanation

{x | x ∈ ℕ, x mod 2 = 0} defines all natural numbers where the remainder when divided by 2 is zero — i.e. all even natural numbers: {2, 4, 6, 8, ...}. This is set comprehension using a mathematical condition. In Python this would be: {x for x in range(1, 101) if x % 2 == 0}. The textbook also shows: {2x | x ∈ ℕ} which generates {2, 4, 6, ...} by doubling each natural number.

How would you write the set of all squares of natural numbers using set comprehension?

• {x | x ∈ ℕ, x = x²}
• {x² | x ∈ ℕ}
• {x | x² ∈ ℕ}
• {x | x ∈ ℕ, x mod 2 = 0}

Answer explanation

{x² | x ∈ ℕ} means 'the set of all x² where x is a natural number'. This generates: {1, 4, 9, 16, 25, ...} (1²=1, 2²=4, 3²=9, etc.). This is a powerful technique — the expression before the | defines what goes INTO the set (x²), while the condition after | defines the source (x ∈ ℕ). In Python: {x**2 for x in range(1, 11)} gives {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}.

What is the empty set, and how is it represented?

• The set {0} — a set containing zero
• A set with no elements, represented by ∅ or {}
• A set containing undefined values
• The set of all negative numbers

Answer explanation

The EMPTY SET is the set with NO elements. It is written as ∅ or {}. Crucially, ∅ ≠ {0} — the empty set contains nothing at all, while {0} contains one element (the number zero). The empty set arises naturally in operations — for example, the intersection of {1,2,3} and {4,5,6} is ∅ because they share no common elements. The empty set has a cardinality of zero.