Homework 1

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Homework 1
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Problems

• 1. Define recursively the set A of all bit
  strings that have as many ones as zeros.
• Ex 10, 1010, 1001, 1110000,
• Counter Ex 0, 001, 100, 10011,
• 2. Define two functions one, zero 0,1 ? N to
  count the number of ones and zeros, respectively,
  of an input bit string.
• ex one(1100) 2 zero(1100) zero(100111) 2.
• 3. Show the correctness of your definition of A
  by proving the followings
• 3.1 for all x in A, one(x) zero(x). use
  structural induction
• 3.2. if x is a bit string which has as many ones
  as zeros then x ? A.

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Exercises

• 4. Define recursively the set B of all bit
  strings that have more ones than zeros. (You may
  use A defined previously though it is not
  necessary. Hint Analyze how a member of B can be
  formed as the concatenation of two members of B
  and possibly some other bits.)
• Ex 1, 110, 101, 11110.
• Counter Ex 0, 10, 001, 100, 0011,
• 5. Show the correctness of your definition of B
  by proving the followings
• 5.1 for all x in B, one(x) gt zero(x). use
  structural induction
• 5.2. if x is a bit string which has more ones
  than zeros then x ? B.

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• 6. The set N of natural numbers is defined as a
  subset of 1 as follows
• Basis e ?N
• closure if x ? N then 1?x ? N.
• Given the domain N, we define the plus function
  as follows
• for all x,y ?N
• basis (x, e) x
• recursion (x, 1y) 1 ? (x,y).
• To convince you that is a correction defintion
  of plus, prove by induction that for all x,y,z
  ? N
• 6.1 is associative i.e., (x, (y,z)) (
  (x,y), z).
• 6.2 is commutative (x,y) (y,x).
• Hint To show the result, you may need to
  establish some lemmas first.

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• 7. Let B be a set of strings over an alphabet S.
  We say B is transitive if BB ? B and reflexive if
  e?B.
• Prove that for any set of strings A, A is the
  smallest reflexive and transitive set containing
  A. That is, show that
• 7.1. A is a reflexive and transitive set
  containing A, and
• 7.2. if B is any other reflexive and transitive
  set containing A, then A ? B.
• Note A is defined to be e U A U A2 U A3 U