Lessons 2-3

• ¬


  Negation of p
• Truth table


  • Illustrates all the possible truth values
  • Example: p and q = 2^2 = 4 rows
  • Example: p, q, and r = 2^3 = 8 rows
• Conjunction


  p and q, true only when both p and q are true
• Disjunction


  p or q, false only when both p and q are false
• Conditional statement


  If p, then q, false only when p is true and q is false
• Converse


  q then p
• Inverse


  ¬p then ¬q
• Contrapositive


  ¬q then ¬p
• Biconditional statement


  p if and only if q, true only if both are true or both are false
• Tautology


  Always true
• Contradiction


  Always false
• Implication


  p logically implies q, if the conditional statement p → q is a tautology
• Equivalence


  p and q are logically equivalent, if they have the same truth table
• De Morgan's Laws


  • Not (A or B) is logically equivalent to Not A and Not B
  • Not (A and B) is logically equivalent to Not A or Not B
• NAND gate


  Equivalent to an OR gate with inverted inputs
• NOR gate


  Equivalent to an AND gate with inverted inputs
• ℕ


  Set of natural or counting numbers (positive integers): {1; 2; 3; 4; …}
• ℤ


  Set of integers: {..., -4; -3; -2; -1; 0; 1; 2; 3; ...}
• ℚ


  Set of rational numbers: {a/b | a, b ∈ ℤ, b ≠ 0}
• ℝ


  Set of real numbers
• x | x


  "x such that x"
• x ∈ ℕ


  "x is a natural number"
• Roster Form


  A = {a, e, i, o, u}

  B = {1, 2, 3, 4, 5}
• Set Builder Notation


  A = {x | x is a vowel of the English alphabet}

  B = {x | x ∈ ℕ, 1 ≤ x ≤ 5}
• Inductive Reasoning


  Arrives at a general conclusion based on the observation of specific examples
• Deductive Reasoning


  Arrives at a specific conclusion based on previously accepted general statements
• George Pólya's Problem Solving Model

  • Understand the Problem

  2. Devise a plan

  3. Carry out the plan

  4. Look back
• Strategies in Pólya's Problem Solving Model

  • Draw a diagram

  2. Solve a simpler problem

  3. Make a table

  4. Work backwards

  5. Guess and check

  6. Find a pattern

  7. Use a formula or an equation

  8. Using logical reasoning