Understanding Math Language and Symbols

Characteristics of Mathematical Language

• Math language does not have past, present, or future tenses
• Math language carries no emotional content and adheres to logic
• Math language is precise, concise, and able to make fine distinctions

Common Mathematical Symbols

• ≠ means not equal to
• = means equals sign
• < means strict inequality

• means strict inequality

• ≤ means inequality
• ≥ means inequality
• [] brackets calculate expression inside first
• () parentheses calculate expression inside first
• • minus sign subtraction
• • plus sign addition

Characteristics of Mathematical Language

• Concise in expression
• Twelve plus twenty eight is equal to forty written as 12 + 28 = 40
• Powerful in expression, requiring critical thinking, comprehension, analysis, and reasoning skills

Mathematical Expression

• A mathematical expression is a group of characters or symbols that represents a quantity or operation
• An algebraic expression includes numbers, variables as letters, and math operations

Mathematical Equation

• A mathematical equation is a statement of equality between two algebraic expressions with unknowns
• There are mathematical sentences, open mathematical sentences, and closed mathematical sentences

PEMDAS/BODMAS

• Use PEMDAS/BODMAS, order of operations to simplify equations:
  • P - Parentheses ()
  • E - Exponents
  • MD - Multiplication x and Division ÷
  • AS - Addition + and Subtraction -

Sets

• A set is a collection of well-defined objects without duplicates
• Objects in a set are called elements
• Sets are described using braces { } and capital letters

Specification of Sets

• Elements or members (€) are objects listed in a set
• The symbol ∈ means the object is found in a set
• The symbol ∉ means "not an element of" a set
• Don't repeat elements and only use capital letters to name a set

Fundamental Number Sets

• Natural Numbers (N): {1, 2, 3, 4, 5, ...}
• Whole Numbers (W): {0, 1, 2, 3, 4, 5, 6, 7, ...}
• Integers (Z): {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}
• Rational Numbers (Q): Terminating or repeating decimals
• Irrational Numbers (QC): Non-terminating, non-repeating decimals
• Real Numbers (R): All rational and irrational numbers

Rule Method / Set-Builder Notation

• {x | x is a natural number and x < 8} = the set of all x such that x is a natural number and is less than 8

Specifications of Sets

• An empty set or null set has no elements: {} or Ø
• A finite set has a countable number of elements
• An infinite set is not finite
• The cardinal number of a finite set A is the number of elements denoted by n(A)

Set Examples

• A finite set has a countable number of elements with a natural number limit
• An infinite set is not finite
• An empty (or null) set has no members: Ø or {}

Equal Sets

• Two sets A and B are equal (A = B) if they have the same elements

Equivalent Sets

• Two sets A and B are equivalent (A ~ B) if they have the same number of elements

Universal Set

• A universal set (U) contains all elements considered in a specific context

Subsets

• Set A is a subset of B if every element of A is also in B

Operations on Sets

• The union of two sets (A ∪ B) contains all elements in either A or B
• The intersection of two sets (A ∩ B) contains elements common to both A and B

Set Complementation

• An operation performed in reference to a universal set, denoted by A'

Disjoint Sets

• Two sets A and B are mutually exclusive or disjoint if they do not have any shared elements; i.e. A∩B=Ø

Relation

• A relation is a rule that pairs each element in one set (domain) with ones from a second set (range), forming ordered pairs

Function

• A function pairs each element in one set (domain) with exactly one element from a second set (range)
• One-to-one and many-to-one correspondences are functions

Representation of functions

• Table
• Ordered Pairs
• Mapping
• Graphing/Vertical line test

Binary Operation

• A binary operation on a set is a calculation involving two elements of the set to produce another element of the set