Lesson 2

• What is the main focus of Lesson 2 in the study material?

  The language, symbols, and conventions of mathematics
• What are the learning objectives at the end of this chapter?

  To discuss mathematical language, explain its nature, perform operations on expressions, and appreciate its usefulness
• What are the characteristics of the language of mathematics?

  • Precise: Makes fine distinctions
  • Concise: Expresses ideas briefly
  • Powerful: Expresses complex thoughts easily
• What does the term "precise" mean in the context of mathematical language?

  It refers to the ability to make very fine distinctions using symbols
• How does mathematical language demonstrate conciseness?

  It allows long English sentences to be shortened using symbols
• What does "powerful" mean in the context of mathematical language?

  It refers to the ability to express complex thoughts with relative ease
• What is a mathematical expression?

  A finite combination of symbols that is well-formed according to rules
• How does a mathematical expression differ from a mathematical sentence?

  An expression does not state a complete thought, while a sentence does
• What is an example of a simple mathematical expression?

  5 + 2
• What is a mathematical sentence or equation?

  An arrangement of symbols that states a complete thought
• What is the role of the verb in a mathematical sentence?

  The verb indicates the relationship between the elements, such as "=" in 3 + 4 = 7
• What is the meaning of the symbol "+" in mathematics?

  Addition
• What does the symbol "∞" represent?

  Infinity
• What is a set in mathematics?

  A collection of well-defined objects that contains no duplicates
• How is a set denoted in mathematics?

  Using braces {} and capital letters
• What is an example of a singleton set?

  A set with only one member, such as {a}
• What is the cardinality of a set?

  The number of distinct elements belonging to a finite set
• What are the different ways to specify a set?

  1. List Notation/Roster Method: Listing all members
  2. Predicate Notation/Rule Method/Set-Builder Notation: Stating a property of its elements
• What is an equal set?

  Two sets that contain exactly the same elements
• What is an equivalent set?

  Two sets that contain the same number of elements
• What is a universal set?

  A set that contains all the elements considered in a particular situation
• What is a subset?


  A set A is a subset of B if every element of A is also an element of B
• What is a proper subset?

  A subset that is not equal to the original set
• What is an improper subset?

  A subset that contains all the elements of the original set
• What is the operation of union in set theory?


  It forms a set that consists of all elements included in A and B or both
• What are the common symbols used in the order of operations?

  • Addition: +
  • Subtraction: -
  • Multiplication: ×
  • Division: ÷ or /
  • Equals: =
  • Pi: π
  • Infinity: ∞
• What are the examples of mathematical sentences or equations?

  1. 3 + 4 = 7
  2. 8x - 7x² + 4x - 10 = 90
• What does the set U represent in the context of whole numbers?

  U represents the set of all whole numbers, denoted as U(0,1,2,3,...)
• What is a subset?


  A set A is called a subset of B if every element of A is also an element of B.
• How is a subset denoted mathematically?

  A is a subset of B is written as A ≤ B.
• Provide an example of a subset.


  • A = {7, 9} is a subset of B = {6, 9, 7}
  • D = {10, 8, 6} is a subset of G = {10, 8, 6}
• What is a proper subset?

  A proper subset is a subset that is not equal to the original set.
• What is an improper subset?

  An improper subset is a subset that contains all the elements of the original set.
• Provide examples of proper and improper subsets.

  • Given {3, 5, 7}:
  • Proper subsets: {}, {5, 7}, {3, 5}, {3, 7}
  • Improper subset: {3, 5, 7}
• What is the cardinality of a set?

  The cardinality of a set is the number of distinct elements belonging to a finite set.
• How is the cardinality of a set denoted?

  The cardinality of set A is denoted by n(A), card(A), or |A|.
• What is the union of two sets?

  • The union of sets A and B consists of all elements included in A and B or both.
  • Denoted by A ∪ B.
• Given U = {1,2,3,4,5,6,7,8,9}, A = {1,3,5,7}, and B = {2,4,6,8}, what is A ∪ B?


  A ∪ B = {1,2,3,4,5,6,7,8}
• Given U = {1,2,3,4,5,6,7,8,9}, A = {1,3,5,7}, and C = {1,2}, what is A ∪ C?


  A ∪ C = {1,2,3,5,7}
• Given U = {1,2,3,4,5,6,7,8,9}, A = {1,3,5,7}, and what is (A ∪ B) ∪ {8}?


  (A ∪ B) ∪ {8} = {1,2,3,4,5,6,7,8}