Homework 5

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Module 9

• • Given that the definition of an odd number is , whereas an even number is , we can prove that the sum of two odd integers is even.
  • Given some odd integer, , and another , the sum is which is of form .
• • Let be some rational number and be some irrational number.
  • Suppose is rational, this means that there exists some such that , where is rational.
  • You can isolate , . Since and are both rational, the difference of two rational numbers is also rational. This would imply that is rational, but that contradicts the original assumption.
  • Since the assumption that is a contradiction, must be irrational.
  • The sum of an irrational number and a rational number is irrational.
• • A rational number is defined as the ratio of two integers, such that , where and are integers, and .
  • Let and be two rational numbers, , and .
  • Multiplying and results in .
  • Since the product of two integers is also an integer, follows the form of , where both and are integers.
  • The product of two rational numbers is always rational.
• • A number is even if it follows the form , and odd if it follows the form .
  • Given that the product of is even, it is of the form .
  • Assuming both are odd, , and .
  • which is odd.
  • Since is odd if both and are odd, then at least one of or must be even for to be even.
• • • The contrapositive is if is odd, then is even.
    • Assuming is odd, , , which is odd.
    • would therefore have to be even, since the sum of two odd numbers is even.
    • By contraposition, must be even give that is odd.
  • • Suppose is odd, and is also odd.
    • Suppose is odd, , which is odd.
    • would be even, but the original statement was that is odd, which is a contradiction.
    • We can conclude that must be even given that is odd.

Module 10

• • • Left Side:
    • Substitute:
    • Factor out :
    • Expand:
    • Factor : which matches right-hand.
  • Since we have proven is true and that if is true, then is true, induction guarantees that the formula holds for all positive integers .
• • • Substitute:
    • Factor out :
    • Expand:
    • Simplify: which matches right-hand.
  • Since we proved is true, and that if is true, then is true, induction guarantees that the formula holds for all positive integers .
• • Base Case ():
  • Inductive Hypothesis:
  • Inductive Step:
    • Substitute:
    • Expand:
    • Factor out :
    • Expand:
    • Factor:
    • Since and , the sides match
  • Since the formula holds for , and that if the formula holds for , then it also holds for , by induction, the formula is true for all positive integers .
• • Base Case ():
  • Inductive Hypothesis:
  • Inductive Step:
    • Substitute:
    • Factor out : , which matches the formula
  • Since the formula holds for , and that if the formula holds for , then it also holds for , by induction, the formula is true for all positive integers .
• • Base Case ():
  • Inductive Hypothesis:
  • Inductive Step:
    • Substitute:
    • Factor out : , which matches the formula
  • Since the formula holds for , and that if the formula holds for , then it also holds for , by induction, the formula is true for all positive integers .
• • Base Case ():
  • Inductive Hypothesis:
  • Inductive Step:
    • Substitute:
    • Factor out : , which matches the formula
  • Since the formula holds for , and that if the formula holds for , then it also holds for , by induction, the formula is true for all positive integers .

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