Worksheet - MATH and the Peano Axioms

  1. Determine if each statement is true or false.

    1. $3\in \QTR{Bbb}{N}.$
      MATH

    2. $3\in \QTR{Bbb}{Q}.$
      MATH

    3. The sum of 2 natural numbers is a natural number.
      MATH

    4. The difference of 2 natural numbers is a natural number.
      MATH

    5. If MATH then MATH
      MATH

    6. If MATH then MATH
      MATH

    7. If MATH then MATH
      MATH

    8. If MATH then MATH
      MATH

  2. For each statement about $\QTR{Bbb}{N}$ below, state which Peano Axioms would need to be used to prove it.

    1. $3\in \QTR{Bbb}{N}.$
      MATH

    2. MATH
      MATH

    3. If MATH and $n+4=m+4,$ then $n=m.$
      MATH

    4. There is no highest natural number.
      MATH

  3. Suppose that 1000 dominoes are lined up so that as soon as one tips over its successor does also.

    1. If I start by tipping over the 1st one, will all the dominoes fall over?
      MATH

    2. If I start by tipping over the 3rd one, will all the dominoes fall over?
      MATH

    3. Do your answers to $a$ and $b$ change if '1000' is replaced by any other natural number?
      MATH

  4. On an alien planet, tipping over a domino means that its successor does not fall, but its successor's successor does. If Zotz the alien lines up 1000 dominoes, what is the easiest way for Zotz to make them all fall?
    MATH

  5. Fill in the blanks in the following proof (don't worry, the proofs will get much more interesting than this very soon.):

      Prove:
      Prove:

      If $n\in \QTR{Bbb}{N},$ then MATH

      Proof:

      Premises: MATH

      Since $n\in \QTR{Bbb}{N},$ we have MATH by $\_\_\_\_$N2$\_\_\_$.

      Because MATH we have MATH by N2.

      It follows, by N2 again, that MATH

      However, from basic arithmetic, MATH

      Conclusion: MATH

  6. Fill in the blanks in the following proof:

      Prove:

      MATH

      Proof:

      Premises: none$.$

      Suppose, for the sake of contradiction, that MATH.

      Then, by $\_\_\_\_$N2$\_\_\_\_,$ the successor of $-1$ is a natural number as well.

      The successor of $-1$ is $\_\_\_0\_\_\_.$

      Thus MATH

      Since $0\in \QTR{Bbb}{N},$ its successor is also a natural number, again by $\_\_\_$N2$\_\_\_$.

      The successor of $0$ is $\_\_\_1\_\_\_.$

      Thus 1 is the successor of a natural number.

      This contradicts $\_\_\_$N3$\_\_\_.$

      Because we have arrived at a contradiction, our supposition that MATH must be false.

      Conclusion: MATH

  7. Prove that there is no highest natural number.

      Proof:
      Proof:

      Premises: none$.$

      Suppose, for the sake of contradiction, that there is a highest natural number.

      Let $n$ be the highest natural number.

      Then its successor, $n+1$, is also a natural number, by N2.

      However, $n+1>n$.

      Thus, we have found a natural number greater than $n.$

      This contradicts the claim above that $n$ is the highest natural number.

      Because we have arrived at a contradiction, our supposition that there is a highest natural number.must be false.

      Conclusion: There is no highest natural number.

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