Let $R$ be the "between the same consecutive powers of 2" relation on $\mathbb{Z}^+$: $aRb \iff \exists k \in \mathbb{Z},(2^k \le a < 2^{k+1}) \wedge (2^k \le b < 2^{k+1})$
a. Explain why $7$ $\not R $ $9$
b. Prove that $R$ is an equivalence relation.
c. List all of the elements of $[5]$
d. Use a particular counterexample to explain why R fails to be an equivalence relation if the definition is adjusted as follows: $aRb \iff \exists k \in \mathbb{Z},(2^k < a < 2^{k+1}) \wedge (2^k < b < 2^{k+1})$
e. Use a particular counterexample to explain why R fails to be an equivalence relation if the definition is adjusted as follows: $aRb \iff \exists k \in \mathbb{Z},(2^k \le a \le 2^{k+1}) \wedge (2^k \le b \le 2^{k+1})$
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I'm particularly confused based on how the question "between the same consecutive powers of 2" for the definition is asked, so I was pretty lost trying to solve any of these. Tips/Answers for even 1 or 2 questions are helpful!! Thanks
What they mean by 'between the same consecutive powers of 2' is that the powers of 2 can be seen as points to split the numberline into different intervals. That is, you have as powers of 2: 1,2,4,8,16, etc., which divides the integers into the intervals [1,2], [2,4], [4,8], etc.
If you look careful at the definition though, you see that the powers of 2 themselves are used as the start of each interval, and not as the end, so the intervals really become [1,2), [2,4), [4,8), etc.
Finally, for each interval we only consider the whole numbers that occur in those intervals, so we get the following groupings:
[1,2) becomes 1 by itself
[2,4) becomes 2 and 3
[4,8) becomes 4,5,6, and 7
Etc.
These groups are the equivalence classes defined by this equivalence relationship: aRb iff a and b are in the same group.
Ok, so this immediately answers the first question: 7 and 9 are not in the same group, so $7 \not R 9$
Oh, and for the second question you have to prove that it is actualy an equivalence relation: for this prove that the relation is reflexive,i.e. That every number is in the same group as itself (of course!), that it is symmetric, i.e. if a and b are in the same group, then b and a are in the same group (of course!), and that it is transitive, i.e. if a and b are in the same group, and b and c are, then a and c are (of course!)
I'll leave the rest to you