Study sheet: Mathematical Language and Sets

📋 Course Outline

1. Characteristics of Mathematical Language
2. Expressions and Mathematical Sentences
3. Translating Mathematical Language
4. Basic Set Concepts
5. Operations on Sets

📖 1. Characteristics of Mathematical Language

🔑 Key Concepts & Definitions

• Mathematical language : a system of conventional spoken, manual, or written symbols used by people to express themselves

📝 Essential Points

• Mathematical language has three stated characteristics:
  • It is non-temporal
  • It carries no emotional content
  • It is precise and concise

💡 Memory Hook

NPC: non-temporal, precise, concise

📖 2. Expressions and Mathematical Sentences

🔑 Key Concepts & Definitions

• Mathematical expression : a group of characters or symbols representing a quantity and/or an operation
• Algebraic expression : a mathematical expression containing numbers, variables represented by letters, and operations such as addition, subtraction, multiplication, or division
• Mathematical sentence : an expression that is either true or false

★ Must-know

📌 An open mathematical sentence may be true or false depending on the values of unknown quantities, whereas a closed mathematical sentence is already known to be true or false.

Further detail

• The sentence 3 + 12 = 15 is a true closed mathematical sentence, while 3 + 12 = 312 is a false closed mathematical sentence.

💡 Memory Hook

Expression represents quantity or operation; sentence is true or false

📖 3. Translating Mathematical Language

★ Must-know

• Words and phrases implying addition include:

  • plus
  • the sum of
  • increased by
  • total
  • more than
  • added to

• Words and phrases implying subtraction include:

  • minus
  • the difference of
  • decreased by
  • fewer than
  • less than
  • subtracted from

Further detail

📌 In English, a sentence expresses a fact or fiction, whereas in mathematics a sentence expresses a true or false statement.

• Words and phrases implying multiplication include:

  • times
  • the product of
  • multiplied by
  • of

• Words and phrases implying division include:

  • divided by
  • the quotient of
  • per

💡 Memory Hook

English uses words and verbs; mathematics uses symbols and operations

📖 4. Basic Set Concepts

🔑 Key Concepts & Definitions

• Set : a collection of common objects, numbers, letters, people, or other sets
• Subset : a set whose every element is found in another set
• Cardinality : Cardinality is the number of elements in a set and is denoted by n of the set.
• Universal set : a set containing all elements found in the other given sets and is denoted by U

★ Must-know

📌 The roster method lists or enumerates the elements of a set, whereas the rule method describes the elements specifically using set-builder notation.

📌 An empty set has no elements, an equal pair of sets has exactly the same elements, and an equivalent pair of sets has the same cardinality.

Further detail

📌 A finite set has a limited number of elements, whereas an infinite set does not have a finite number of elements.

💡 Memory Hook

A universal set as a large container holding smaller sets

📖 5. Operations on Sets

🔑 Key Concepts & Definitions

• Set union : The union of two sets contains all elements that are in either set, and union is commutative because A union B equals B union A.
• Set intersection : The intersection of two sets contains all elements common to both sets, and intersection is commutative because A intersection B equals B intersection A.
• Set complement : The complement of set A contains all elements x in the universal set U such that x is not in A.
• Set difference : The difference of A and B contains all elements that are in A and not in B.
• Symmetric difference : The symmetric difference of A and B contains elements belonging to A or B but not to both.
• Cartesian product : The Cartesian product A times B consists of ordered pairs formed by pairing each element of A with each element of B, with the A element written first.

💡 Memory Hook

Union, intersection, complement, difference, symmetric difference, Cartesian product

📊 Synthesis Tables

English and Mathematical Languages

Dimension	English	Mathematics
Symbols	English alphabet and punctuation	English alphabet, numerals, Greek letters, grouping symbols, and special symbols
Names	Nouns and pronouns	Expressions and variables
Action	Verbs	Operations and other actions
Sentence value	Fact or fiction	True or false