Please answer the following induction problems.

- Use induction to solve the following problem.

**P: An integer of the form 7 ^{N}-1 is divisible by 6 for all positive integers N**

- Use induction to solve the following problem.

**P: A set which contains N elements has exactly 2 ^{N} subsets.**

- Use induction to prove that a tree with N levels has at most 2
^{N}– 1 nodes. Note that the base cases are 1-levels ( meaning at most one node – the root), and 2-level has at most 3 nodes.

- What
is wrong with the following strong induction “proof” that all horses in a field must be of the same color.*exactly*

Induction on the number of horses in a field.

Base case : P(1) : Trivially, if a field has one horse in it then all horses in the field are of the same color.

Inductive Hypothesis P(k) : ASSUME that all horses in a field of K horses are the same color.

Show P(k+1) : Show that in a field with k+1 horses, all horses are the same color.

Remove a horse X from the field leaving k horses. By applying the inductive hypothesis to the remaining k horses we know that the remaining horses are all the same color.

Next, remove a DIFFERENT horse Y from the field. Again, by the inductive hypothesis, since there are k horses remaining then all of the horses in the field are the same color. Thus, since X != Y we can deduce that all k+1 horses are the same color.

**Languages Quiz:**

*Draw the transition diagram (the graph) for a DFA accepting the following languages over the alphabet {0,1}.*

- L = {w | the 4th symbol from the end of w is a 1}

- L = {w | the # of 0’s in w is divisible by 3 and the # of 1’s is divisible by 2} Hint (look at remainders when divided by 3 for # of 0’s or by 2 for # of 1’s)

- L = { w | w is a string of 0’s and 1’s and when interpreted as a binary number (interpretted as base-10) w is an
number}__even__

- L = { w | w begins with the substring 0010
the substring 0110 }__or__

- L = {w | w does
end with 0001 }__not__

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