Chapter 2

• Relation Symbols

  Greater than or equal (≥), less than or equal (≤), equal (=), not equal (≠), similar (~), approximately equal (≈), and congruent (≅)
• Elements of the set of numbers

  • Real numbers (ℝ)
  • Rational numbers (ℚ)
  • Irrational numbers(ℚ’)
  • Integers(ℤ)
  • Natural numbers (ℕ)
• Dr. Burns: '“the language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express”'
• Classification of Symbols

  • Numbers
  • Operation Symbols
  • Relation Symbols
  • Grouping Symbols
  • Variables
  • Set theory symbols
• Grouping Symbols

  Parentheses ( ), curly brackets or braces { }, or square brackets [ ]
• Set theory symbols

  • subset (⊆), union(∪), intersection(∩), element(∈), not element(∉), empty set (∅)
• Number


  A mathematical object used to count, quantify, and label another object
• Operation Symbols

  Addition (+), subtraction (-), multiplication (× or ∙), division (÷ or/), and exponentiation (𝑥𝑛), where 𝑥 is the base and 𝑛 is the exponent
• Language


  A systematic means of communicating ideas or feelings by the use of conventionalized signs, sounds, gestures, or marks having understood meanings
• Characteristics of language

  • Precise - able to make very fine distinctions
  • Concise - able to say things brief
  • Powerful - able to express complex thoughts with relative ease
• Variables


  Another form of mathematical symbol used when quantities take different values, usually include letters of the alphabet like 𝒙, 𝒚, 𝒛, 𝒂, 𝒃, 𝒄
• Congruent figures are the same shape and size. Similar figures are the same shape, but not necessarily the same size. Two quantities are approximately equal when they are close enough in value so the difference is insignificant in practical terms. For example, 4.9999999 ≈ 5
• Logic symbols

  • Implies (⇒)
  • Equivalent (⇔)
  • And (∧)
  • Or (∨)
  • For all (∀)
  • There exists (∃)
  • Therefore (∴)
• Subset symbols used in the study of sets
• Types of Sentences

  • Open Sentence – is a sentence that uses variables; thus it is not known whether or not the mathematical sentence is true or false.
  • Close Sentence – is a mathematical sentence that is known to be either true or false.
• Mathematical Expression

  An expression is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. An expression does not state a complete thought.
• Mathematical symbols for quantities taking different values

  • 𝒙, 𝒚, 𝒛, 𝒂, 𝒃, 𝒄
• Statistical symbols

  • sample mean ( ҧ𝑥)
  • population mean (𝜇)
  • median (෤𝑥)
  • population standard deviation (𝜎)
  • summation (σ )
  • factorial (n!)
• Set theory symbols

  • subset (⊆)
  • union(∪)
  • intersection(∩)
  • element(∈)
  • not element(∉)
  • empty set (∅)
• Mathematical Sentence

  A mathematical sentence is the analogue of an English sentence; It is a correct arrangement of mathematical symbols that expresses a complete thought.
• Universal Set


  A set containing all the elements under consideration, denoted by ��
• Empty Set or Null Set


  The set with no elements, denoted by ∅ or { }
• Set


  A well-defined collection of distinct objects
• Any set is a subset of itself
• Subsets


  A set 𝐴 is a subset of a set 𝐵 if every element of 𝐴 is also an element of 𝐵
• Equal Sets


  Two sets are equal if they have exactly the same elements
• List all the subsets of a set

  Enumerate all possible subsets of a given set
• Union of sets


  The set containing all elements which belong to either of the sets or both
• A set with 𝑛 elements has a total of 2𝑛 subsets
• The null set is a proper subset of every set
• Singleton Set

  The set with only one element
• Finite and Infinite set


  A set is finite if it consists of a finite number of elements; otherwise, it is infinite
• Element of a set

  Each object belonging to a set
• Equivalent Sets


  If two sets have the same number of elements, they are considered equivalent sets
• The complement of a set A, written A′, is the set of all elements which are in the universal set U but not in A
• The union of A and B, written A∪B, is the set containing all the elements which belong to either A or B or to both
• has a total of 2�� subsets
• Function
• In mathematics, a relation R from set X to set Y is a subset of X × Y
• Relation