lesson 2

• Mathematical language and symbols


  Systematic way of communication with other people using sounds or convention symbols
• Importance of language


  • Language was invented to communicate ideas with others
  • The language of mathematics was designed for numbers, sets, functions, and performing operations
• Symbols commonly used in mathematics

  • The ten digits
  • Operations
  • Sets
  • Variables
  • Special symbols
  • Logic symbols
  • Set notations
• Some important sets


  • N= {1,2,3,...} (natural numbers)
  • W= {0,1,2,3,...} (whole numbers)
  • Z= {...,-3,-2,-1,0,1,2,3,...} (integers)
  • Q (rational numbers)
  • Q' (irrational numbers)
  • R (real numbers)
• Grammar of mathematics


  The mathematical notation used for formulas has its own grammar, not dependent on a specific natural language, but shared internationally by mathematicians regardless of their mother tongues
• Characteristics of mathematics language

  • Precise - able to make very fine distinctions
  • Concise - able to say things briefly
  • Powerful - able to express complex thoughts with relative ease
• The word "is" could mean equality, inequality, or member in a set
• Different use of a number
• Mathematical objects may be represented in many ways such as sets and functions
• The word "and" and "or" mean differently in math from its English use
• Variables


  A symbol for a value we don't know yet
• Advantage of using variables

  • It allows you to give a temporary name to what you are seeking so that you can perform concrete computations with it to help discover its possible values
• Using variables


  • Is there a number with the property that doubling it and adding 3 gives the same result as squaring it?
  • No matter what number might be chosen, if it is greater than 2, then its square is greater than 4
• Writing sentences using variables

  1. Are there numbers with property that the sum of their squares equals the square of their sum?

  2. Give any real number, it square is nonnegative
• Universal statement


  Says that a certain property is true for all elements in a set. "for all"
• Conditional statement


  Says if one thing is true then some other thing also has to be true. "if-then"
• Existential statement


  Says that there is at least one thing for which the property is true
• Universal conditional statement


  A statement that is both universal and conditional
• Rewriting a universal conditional statement

  • For all real number x, if x is nonzero then x^2 is positive
• Universal existential statement


  A statement that is universal because its first part is that a certain property is true for all objects of a given type, and it is existential because its second part asserts the existence of something
• Rewriting a universal existential statement

  • Every pot has a lid
  • All pots _______
  • For all pots p, there is _____
  • For all pots P, there is a lid L such that _____
• Existential universal statement


  A statement that is existential because its first asserts that a certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind
• Rewriting existential universal statement

  • There is a person in my class who is at least as old as every person in my class
  • Some ___ is at least as old as _____
  • There is a person P in my class that P is ____
  • There is a person P in my class with the property that for every person q in my class, p is _____
• Sets


  A well-defined collection of distinct objects, usually represented by capital letters with elements separated by commas
• If S is a set, the notation X S means that x is an element of S
• The symbol ... is called an ellipses and is read "and so forth"
• Some important sets


  • N (natural numbers)
  • W (whole numbers)
  • Z (integers)
  • Q (rational numbers)
  • Q' (irrational numbers)
  • R (real numbers)
  • C (complex numbers)
• Set-builder notation


  Let S denote and P(x) be a property that element of S may or may not satisfy. We define a new set to be the set of all elements x in S such that P(x) is true
• Using the set-builder notation


  • Given that R the set of all real numbers, Z the set of all integers, and Z+ the set of all positive integers, describe the following sets
• Finite set


  A set with a countable number of elements
• Infinite set


  A set with an uncountable number of elements
• Equal sets


  Sets with exactly the same elements and cardinality
• Equivalent sets


  Sets with the same number of elements or cardinality
• Joint sets


  Sets with common elements (intersection)
• Disjoint sets


  Sets with no common elements