Problem Solutions

page 73:

Problem:

Characterize the following set of integers (or its negative space):

1, 3, 7, 12, 18, 26, 35, 45, 56, 69, ...

Define F to be the given set, and G to be its complement:

F = {1, 3, 7, 12, 18, 26, 35, 45, 56, 69, ...}

G = {2, 4, 6, 8, 9, 10, 11, 13, ...}

The sets F and G are implicitly defined by the following rules:

1. f₁ = 1
2. f_n+1 - f_n = g_n
3. Sets F and G partition the positive integers (that is, they are disjoint sets whose union is the positive integers)

page 212:

Problem:

Translate the following wfs into English and determine if they are true or false:

1. "~Ac:Eb:(SS0*b)=c"
2. "Ac:~Eb:(SS0*b)=c"
3. "Ac:Eb:~(SS0*b)=c"
4. "~Eb:Ac:(SS0*b)=c"
5. "Eb:~Ac:(SS0*b)=c"
6. "Eb:Ac:~(SS0*b)=c"

Solution:

1. Not all numbers are even. (true)
2. All numbers are not even. (false)
3. For every number, there is an even number that is not equal to it. (true)
4. There is no even number that is equal to all numbers. (true)
5. There is an even number such that not all numbers are equal to it. (true)
6. There is an even number that is not equal to any number. (false)

page 215:

Problem:

Translate the following sentences into TNT:

1. All natural numbers are equal to four.
2. There is no natural number which equals its own square.
3. Different natural numbers have different successors.
4. If 1 equals 0, then every number is odd.
5. b is a power of 10.

Solution:

1. "Aa:a=SSSS0"
2. "~Ea:a=(a*a)"
3. "Aa:Ab:<~a=b}~Sa=Sb>"
4. "<S0=0}Aa:Eb:a=S(b+b)>"
5. The solution to this problem is rather involved, so I have described it on a separate page.

page 220 (top):

Problem:

Consider the following incorrect "derivation" of the wf "Ax:a=a":

1.  Aa:(a+0)=a                   axiom 2
2.  Ax:a=(a+0)                   symmetry
3.  Aa:a=a                       transitivity (lines 2, 1)

Solution:

The applications of symmetry and transitivity in lines (2) and (3) are incorrect. A correct derivation is

1.  Aa:(a+0)=a                   axiom 2
2.  (a+0)=a                      specification
3.  a=(a+0)                      symmetry
4.  a=a                          transitivity (lines 2, 3)
5.  Aa:a=a                       generalization

page 220 (bottom):

Problem:

Translate Peano's fourth postulate into TNT, and then derive that wf as a theorem.

Solution:

A translation of Peano's fourth postulate is "Aa:Ab:<~a=b}~Sa=Sb>". Here is a derivation of this wf:

1.  [                           push
2.    Sa=Sb                     premise
3.    a=b                       drop S
4.  ]                           pop
5.  <Sa=Sb}a=b>                 fantasy
6.  <~a=b}~Sa=Sb>               contrapositive
7.  Ab:<~a=b}~Sa=Sb>            generalization
8.  Aa:Ab:<~a=b}~Sa=Sb>         generalization

page 512:

Problem:

Given the rules of typogenetics, find a self-replicating strand.

Solution:

Some examples self-replicating strands are given on a separate page about typogenetics.

page 564:

Problem:

Find the smallest number expressible as the sum of two squares in two different ways.

Solution:

The smallest number expressible as the sum of two squares in two different ways is 50:

50 = 5² + 5² = 7² + 1².