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Induction Problems

Please answer the following induction problems.

1) Use induction to solve the following problem. 
P: An integer of the form 7N-1 is divisible by 6 for all positive integers N

2) Use induction to solve the following problem.
P: A set which contains N elements has exactly 2N subsets.

3) Use induction to prove that a tree with N levels has at most 2N – 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.

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

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}.

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

2. 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)

3. 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 even number}

4. L = { w | w begins with the substring 0010 or the substring 0110 }

5. L = {w | w does not end with 0001 }

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