Mathematics in the Modern World

• Inductive reasoning


  Drawing a general conclusion from a repeated observation or limited set of observations
• Inductive reasoning


  • The conclusion may be true or false depending on the truthfulness of the argument
  • One counterexample can disprove the conclusion
• Deductive reasoning


  • The conclusion necessarily follows from the premises if the premises are true
  • The conclusion cannot be false if the premises are true
• Intuition carries a heavy load of mystery and ambiguity and is somewhat dangerous, illegitimate substitute for a formal proof
• Proof


  A deductive argument for a mathematical statement, demonstrating that the statement is always true
• In mathematical arguments, statements can only be used if they are already proven. Axioms may be served as conditions that must be met before the statements applies
• An unproved proposition that is believed to be true is known as a conjecture
• Inductive Reasoning is a process that makes generalization based on the pattern observed on specific examples.
• Probability is a measure of how likely something is to happen
• Mathematical Induction is an important technique for proving properties of natural numbers
• Principle of Mathematical Induction states that if P(n) is a property of all positive integers n greater than or equal to some fixed number k, then P(k) is true and whenever P(m) is true, it follows that P(m+1) is also true.
• The conclusion formed is called a conjecture.
• We conjecture that the sum of two odd numbers is always even.
• We conjecture that the product of an odd number and an even number is always even.
• If we find just one case which nullifies our conjecture, we call it a counterexample.
• The Tower of Hanoi is a puzzle invented by Edouard Lucas in 1883.
• Tower of Hanoi - The puzzle consists of three pegs and a number of disks of distinct diameters stacked on one of the pegs such that the largest disk is on the bottom, the next largest is placed on the largest disk
• The goal of the puzzle is to move the tower to the rightmost peg adhering to the following rules:

  • Move only one disk at a time.
  • A larger disk can not be placed on top of a smaller disk.
  • All disks, except the one being moved, must be on a peg.
• Deductive Reasoning is a process that makes a specific conclusion based on general assumptions, procedures, facts, rules, or principles.
• Sudoku is a deductive reasoning, number-placement puzzle.
• Inductive Reasoning

  example: During the past 10 years, a tree has produced plum every other year. Last year the tree did not produce plums, so this year the tree will produce plums.
• Logic puzzles can be answered using deductive reasoning and charts that enable individuals to display given information in a visual form (Aufmann et al. 2003).
• George Polya (1887-1985), Hungarian mathematician, proposed a basic problem solving strategy which is composed of four steps.
• Deductive Reasoning
• Deductive Reasoning 


  Example: All triangles have exactly three sides. Figure A is a triangle. Therefore, Figure A has exactly three sides.
• My sister receives her pay check
• Inductive Reasoning 


  Example: My sister receives her pay-check every other Friday. She did not receive her pay last Friday. Therefore, she will receive it this Friday.
• N


  Use inductive reasoning to predict the next letter in the series: O,T,T,F,F,S,S,E,...
• The product of any two consecutive counting numbers is always even.
• The product of any two consecutive counting numbers is always even.
• The sum of any three consecutive whole number is always a multiple of three.
• $X=$$1$


  For all numbers $x, 2x > x.$The following are counterexamples of the given mathematical sentence above EXCEPT
• $x=$$-1$


  For all numbers $x, x+$$1/x+$$1 =$$1$. Which of the following is a counterexample of the given mathematical sentence above?
• $305$


  What is the next term in the sequence? $-5, 10, 29, 60, 111, 190,...$
• $6$


  What is the units digit in the sum: $1²⁰⁰ +$$5⁸⁰⁰$?
• 20 ducks and 15 pigs


  There are ducks and pigs in a farm. Together, there are 35 heads and 100 pigs. Assuming that each duck has exactly two legs and each pig has exactly four legs. How many are the ducks and the pigs in the field?
• $49$


  Use inductive reasoning to predict the next number in the series: $1,4,9,16,25,36,...$
• 105 handshakes


  If 15 people greet each other at a meeting by shaking hands with one another, how many handshakes take place?