1.
We will say that two numbers between 1 and 31 are related, written \(a \sim b\) if their difference is a multiple of 7. So for example, \(3 \sim 24\text{,}\) since \(24-3 = 3\cdot 7\text{,}\) but \(3 \not\sim 15\) since \(15-3 = 12\) which is not a multiple of 7.
(a)
Which of the following are true? That is, which of the following pairs of numbers are related as we have defined above?
- \(\displaystyle 4\sim 14\)
- \(\displaystyle 7\sim 14\)
- \(\displaystyle 10 \sim 17\)
- \(\displaystyle 17\sim 24\)
- \(\displaystyle 10\sim 24\)
- \(\displaystyle 20 \sim 10\)
- \(\displaystyle 31 \sim 3\)
- \(\displaystyle 25 \sim 25\)
(b)
Which of the following statements are true about the \(\sim\) relation in this case?
- \(a \sim a\) for every number \(a\)
- \(a \not\sim a\) for any number \(a\)
- For any numbers \(a\) and \(b\text{,}\) if \(a \sim b\text{,}\) then \(b \sim a\)
- For any numbers \(a\) and \(b\text{,}\) if \(a \sim b\) and \(b \sim a\text{,}\) then \(a = b\)
- For any numbers \(a\) and \(b\text{,}\) if \(a \sim b\) and \(b \sim c\text{,}\) then \(a \sim c\)
(c)
We will write \([a]\) for the set of all numbers related to \(a\text{.}\) For example, \([7] = \{7, 14, 21, 28\}\text{.}\) Find each of the following:
- \([1] =\) ;
- \([2] =\) ;
- \([3] =\) ;
- \([4] =\) ;
- \([5] =\) ;
- \([6] =\) .
Are there any numbers that are in more than one of the sets \([a]\) above?
- Yes
- No
