1. (6 points) Give recursive definitions of each of the following sets. Remember
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Question
1. (6 points) Give recursive definitions of each of the following sets. Remember to include a basis step and a recursive step. Give a brief justification of why your definition works (a) The set of odd integers (both positive and negative) (b) The set of coordinates on the integer grid Z × Z where the sum of the coordinates is even. This erample might help you get started: Let's give a recursive definition of the set of positive multiples of 3. We'll call the set S Basis step: 3ES Recursive step: If x, y are both in S then x+yES The positive multiples of 3 are 3,6,9,12,. We can build each one from earlier ones by adding 3 over and over agamExplanation / Answer
(a)
Definition of odd integers be defined as
BASE: 1 is an odd integer.
RECURSION: If k is an odd integer, then 2k+1 is an od integer.
Now 1 is an odd integer by the definition base.
with k=1, 2k+1=3, so 3 is an odd inteer.
with k=-1, 2k+1=-1, so -1 is an odd integer.
and so ...,-3,-1,1,3,... are odd integers.
(b)
The grid has a pair of coordinates (x,y)
BASE: x=1, y=1, sum of coordinates x+y=2 is even.
RECURSION: If x=k, y=k, then Sum of coordinates x+y=k+k=2k is even.
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