Maths for regular expressions: Be familiar with the concept of a set and the following notations for specifying a set: A = {1, 2, 3, 4, 5 } or set comprehension: A = {x | x ∈ ℕ ∧ x ≥ 1 } where A is the set consisting of those objects x such that x ∈ ℕ and x ≥ 1 is true. Know that the empty set, {}, is the set with no elements. Know that an alternative symbol for the empty set is Ø. A set is an unordered collection of values in which each value occurs at most once. Several languages support set construction. In Python, for example, use of curly braces constructs a set: {1, 2, 3 }. | means such that. x ∈ ℕ means that x is a member of the set ℕ consisting of the natural numbers, ie {0, 1, 2, 3, 4, … }. The symbol ∧ means AND. The term ∧ x > = 1 means AND x is greater than or equal to 1. In Python, {2 ∗ x for x in {1, 2, 3 }} constructs {2, 4, 6 }. This is said to be a set comprehension over the set {1, 2, 3 }. Be familiar with the compact representation of a set, for example, the set {0n1n | n ≥ 1}. This set contains all strings with an equal number of 0 s and 1s. For example, {0n1n | n ≥ 1} = {01, 0011, 000111, 00001111, … } Be familiar with the concept of: • finite sets • infinite sets • countably infinite sets • cardinality of a finite set • Cartesian product of sets. A finite set is one whose elements can be counted off by natural numbers up to a particular number, for example as: 1st element, 2nd element, …, 20th (and final) element. The set of natural numbers, ℕ and the set of real numbers, ℝ are examples of infinite sets. A countably infinite set is one that can be counted off by the natural numbers. The set of real numbers is not countable. The cardinality of a finite set is the number of elements in a set. Cartesian product of two sets, X and Y, written X x Y and read 'X cross Y', is the set of all ordered pairs (a, b) where a is a member of A and b is a member of B. 58 Be familiar with the meaning of the term: • subset • proper subset • countable set. {0, 1, 2 } ⊂ ℕ where ⊂ means proper subset of, that is ℕ contains everything in {0, 1, 2 } but there is at least one element in ℕ that is not in {0, 1, 2 }. {0, 1, 2 } ⊆ {0, 1, 2, 3 } where ⊆ means subset of. ⊆ includes both ⊂ and =, for example {0, 1, 2, 3 } ⊆ {0, 1, 2, 3 } is also true, because {0, 1, 2, 3 } = {0, 1, 2, 3 }. A countable set is a set with the same cardinality (number of elements) as some subset of natural numbers. Be familiar with the set operations: • membership • union • intersection • difference. The set difference A\B (or alternatively A-B) is defined by A\B = {x: x ∈ A and x ∉ B}

Quick explanation

Maths for regular expressions: Be familiar with the concept of a set and the following notations for specifying a set: A = {1, 2, 3, 4, 5 } or set comprehension: A = {x | x ∈ ℕ ∧ x ≥ 1 } where A is the set consisting of those objects x such that x ∈ ℕ and x ≥ 1 is true. Know that the empty set, {}, is the set with no elements. Know that an alternative symbol for the empty set is Ø. A set is an unordered collection of values in which each value occurs at most once. Several languages support set construction. In Python, for example, use of curly braces constructs a set: {1, 2, 3 }. | means such that. x ∈ ℕ means that x is a member of the set ℕ consisting of the natural numbers, ie {0, 1, 2, 3, 4, … }. The symbol ∧ means AND. The term ∧ x > = 1 means AND x is greater than or equal to 1. In Python, {2 ∗ x for x in {1, 2, 3 }} constructs {2, 4, 6 }. This is said to be a set comprehension over the set {1, 2, 3 }. Be familiar with the compact representation of a set, for example, the set {0n1n | n ≥ 1}. This set contains all strings with an equal number of 0 s and 1s. For example, {0n1n | n ≥ 1} = {01, 0011, 000111, 00001111, … } Be familiar with the concept of: • finite sets • infinite sets • countably infinite sets • cardinality of a finite set • Cartesian product of sets. A finite set is one whose elements can be counted off by natural numbers up to a particular number, for example as: 1st element, 2nd element, …, 20th (and final) element. The set of natural numbers, ℕ and the set of real numbers, ℝ are examples of infinite sets. A countably infinite set is one that can be counted off by the natural numbers. The set of real numbers is not countable. The cardinality of a finite set is the number of elements in a set. Cartesian product of two sets, X and Y, written X x Y and read 'X cross Y', is the set of all ordered pairs (a, b) where a is a member of A and b is a member of B. 58 Be familiar with the meaning of the term: • subset • proper subset • countable set. {0, 1, 2 } ⊂ ℕ where ⊂ means proper subset of, that is ℕ contains everything in {0, 1, 2 } but there is at least one element in ℕ that is not in {0, 1, 2 }. {0, 1, 2 } ⊆ {0, 1, 2, 3 } where ⊆ means subset of. ⊆ includes both ⊂ and =, for example {0, 1, 2, 3 } ⊆ {0, 1, 2, 3 } is also true, because {0, 1, 2, 3 } = {0, 1, 2, 3 }. A countable set is a set with the same cardinality (number of elements) as some subset of natural numbers. Be familiar with the set operations: • membership • union • intersection • difference. The set difference A\B (or alternatively A-B) is defined by A\B = {x: x ∈ A and x ∉ B}

• This point belongs to Regular languages, especially Maths for regular expressions.
• You need to be able to maths for regular expressions: Be familiar with the concept of a set and the following notations for specifying a set: A = {1, 2, 3, 4, 5 } or set comprehension: A = {x | x ∈ ℕ ∧ x ≥ 1 } where A is the set consisting of those objects x such that x ∈ ℕ and x ≥ 1 is true. Know that the empty set, {}, is the set with no elements. Know that an alternative symbol for the empty set is Ø. A set is an unordered collection of values in which each value occurs at most once. Several languages support set construction. In Python, for example, use of curly braces constructs a set: {1, 2, 3 }. | means such that. x ∈ ℕ means that x is a member of the set ℕ consisting of the natural numbers, ie {0, 1, 2, 3, 4, … }. The symbol ∧ means AND. The term ∧ x > = 1 means AND x is greater than or equal to 1. In Python, {2 ∗ x for x in {1, 2, 3 }} constructs {2, 4, 6 }. This is said to be a set comprehension over the set {1, 2, 3 }. Be familiar with the compact representation of a set, for example, the set {0n1n | n ≥ 1}. This set contains all strings with an equal number of 0 s and 1s. For example, {0n1n | n ≥ 1} = {01, 0011, 000111, 00001111, … } Be familiar with the concept of: • finite sets • infinite sets • countably infinite sets • cardinality of a finite set • Cartesian product of sets. A finite set is one whose elements can be counted off by natural numbers up to a particular number, for example as: 1st element, 2nd element, …, 20th (and final) element. The set of natural numbers, ℕ and the set of real numbers, ℝ are examples of infinite sets. A countably infinite set is one that can be counted off by the natural numbers. The set of real numbers is not countable. The cardinality of a finite set is the number of elements in a set. Cartesian product of two sets, X and Y, written X x Y and read 'X cross Y', is the set of all ordered pairs (a, b) where a is a member of A and b is a member of B. 58 Be familiar with the meaning of the term: • subset • proper subset • countable set. {0, 1, 2 } ⊂ ℕ where ⊂ means proper subset of, that is ℕ contains everything in {0, 1, 2 } but there is at least one element in ℕ that is not in {0, 1, 2 }. {0, 1, 2 } ⊆ {0, 1, 2, 3 } where ⊆ means subset of. ⊆ includes both ⊂ and =, for example {0, 1, 2, 3 } ⊆ {0, 1, 2, 3 } is also true, because {0, 1, 2, 3 } = {0, 1, 2, 3 }. A countable set is a set with the same cardinality (number of elements) as some subset of natural numbers. Be familiar with the set operations: • membership • union • intersection • difference. The set difference A\B (or alternatively A-B) is defined by A\B = {x: x ∈ A and x ∉ B}.
• Use the linked flashcards and practice questions to check recall, then practise applying the idea in an exam-style answer.