Long Exams

• Public Attitudes and Perceptions of Math: It is a universal phenomenon that students find mathematics less and less attractive to study. They go from enjoying it much in their preschool days, to actually abhorring it by the time they finish high school.
• Negative attitudes toward math needs to change, especially in a world that is increasingly dependent on high technologies that are brought about by math-related knowhow, among them gadgetry, communication, transportation, and a vast variety of transactions.
• Mathematics is ubiquitous in the most "lowtech" and natural of settings. There is math in sports, arts, and the weather.
• Numeracy and Quantitative Literacy

  • Have a broad general awareness of mathematics, its role in society, and its strengths and limitations
  • Have a working competency in quantitative reasoning – i.e. a good sense of number, shape and size, the ability to assess numerical, statistical and probabilistic evidence, and a level of comfort with abstractly presented reasoning
• It is a popular belief that mathematics is all about computation. Hence, it is important to have an appreciation of, first of all, the need for numbers.
• Dr. James Bullock (1994): 'Mathematics is not a way of hanging numbers on real things. It is a language that allows one to think about extraordinary questions... Getting the picture does not mean writing the formula or crunching the numbers, it means grasping the metaphor.'
• Kelly Gallagher (2004): 'When you empower your students to compare their before and after actions when solving problems..., you allow them not only to see the big picture but to grasp the mathematical metaphor. Teaching students to think metaphorically sharpens their interpretative skills and helps them reach deeper understanding.'
• Logic is the foundation on which mathematics is built. We apply logic to deduce properties of these objects and rules based on some axioms.
• Mathematical Statement


  A (declarative) statement that can be assigned a truth value and classified as true or false, but not both
• Mathematical Statements


  • 1 + 1 = 2
  • 2 + 3 = 6
  • All roses are red
  • The Philippines has more than 17,000 islands
• Not Mathematical Statements


  • Happy Birthday!
  • Message me
  • Can we be friends?
  • 5 + 1
  • x + 3 = 0
  • Mathematics is interesting.
  • 7 is a lucky number.
• Logical Connectives


  • Conjunction/And (∧)
  • Disjunction/Or (∨)
  • Conditional/If then (⇒)
  • Biconditional/If and only if (⇔)
  • Negation/Not (∼)
• Conjunction (of p and q)


  1. True if both p and q are true

  2. False otherwise
• Disjunction (of p and q)

  1. True if at least one of p or q is true

  2. False only if both p and q are false
• Conditional statement (p implies q)

  1. False only when the premise p is true and the conclusion q is false

  2. Otherwise, it is true
• Biconditional statement (p if and only if q)


  1. True if p and q have the same truth value (both true or both false)

  2. False otherwise
• Negation (of p)


  1. True precisely when p is false

  2. False if p is true
• There are 2^n rows for every n number of statements in a truth table.
• Implication


  If statement p (materially) implies statement q, we denote this by "p ⇒ q" (read as p implies q)
• Equivalence


  We denote the equivalence of two statements p and q by p ⇔ q
• Negation of Simple Statements

  ~p: 1 + 1 ≠ 2

  ~q: 2 + 3 ≠ 6

  ~r: Not all roses are red

  ~s: The Philippines has at most 17,000 islands
• Negation of Compound Statements


  ~(p ∧ q) ⇔ ~p ∨ ~q

  ~(p ∨ q) ⇔ ~p ∧ ~q
• Negation of Statements with Quantifiers

  ~p: Not all roses are red

  ~p: Some roses are not red

  ~q: No roses are red

  ~r: Not all roses are red
• Equivalent Forms of Conditional Statements

  • q if p

  q is necessary for p

  p only if q

  All p are q

  p is sufficient for q

  Either not p or q
• Converse (of p ⇒ q)

  The converse of the compound statement is q ⇒ p
• Inverse (of p ⇒ q)

  The inverse of the compound statement is ~p ⇒ ~q
• Contrapositive (of p ⇒ q)


  The contrapositive of the compound statement is ~q ⇒ ~p
• Negation, Disjunction, Conjunction, Conditional, Biconditional

  • Negation: ~p: 5 is not a prime number

  Disjunction: p ∨ q: 5 is a prime number or 5 times 9 is equal to 46

  Conjunction: p ∧ q: 5 is a prime number and 5 times 9 is equal to 46

  Conditional: p ⇒ q: If Maria learns discrete mathematics, then she will find a good job

  Biconditional: p ⇔ q: 3 is odd if and only if 4 is even
• An argument is invalid if there can be two or more possible Euler Diagrams for a statement. An argument is valid if there can only be one Euler Diagram for a statement.
• Modus ponens


  (p ⇒ q) ∧ p ⇒ q
• Modus tollens


  (p ⇒ q) ∧ ~q ⇒ ~p
• Syllogism


  (p ⇒ q) ∧ (q ⇒ r) ⇒ (p ⇒ r)
• Fallacies are arguments given that are not tautologies since they are not true for each of the four cases.
• Fallacies


  • Fallacy of the Converse: [(p ⇒ q) ∧ q] ⇒ p

  Fallacy of the Inverse: [(p ⇒ q) ∧ ~p] ⇒ ~q

  Ad Hominem, Ad Populum, Appeal to Authority, False Cause, Hasty Generalization
• Deductive Reasoning


  The process of reasoning from a general statement to a specific instance
• Inductive Reasoning


  The process of reasoning from specific instances to a general statement
• The study of axiomatic systems can be traced back to the Greeks, who as early as 600 B.C. began to study the logical connections among mathematical facts.
• Euclid organized most of the known mathematics of his time so that virtually all theorems were proved from a small collection of definitions and axioms.
• Axiomatic Method


  The distinctive structure of mathematics (and much of science)
• Components of an Axiomatic System

  • Defined Terms

  Undefined Terms

  Axioms

  Theorems