Counting the elements in sets

[lemonthree]

Question: How many elements are in each set?

For the first set, I think it's 8995 because the set is the union of {1,2,3,4,5},{1,2,3,4,5,6},...{1,2,3,...9000}. So 9000 - 5 = 8995.

For the second set, I'm not too sure about counting the elements in the set. Since \(\displaystyle 1<x≤i\), I can't think of any x mod i = 2.
For example, I know 5 mod 3 = 2, but 5 > 3 and in this case it wants i to be greater or equal to x...any hints please?

 

[HOI]

There are, of course, 9000 integers from 1to 9000. Why are you subtracting 5? Which integers are missing?

A number, n, is congruent to 2 (mod i) if n= i+ 2. Every number, except 1 and 2, is equal to i+ 2 for some i.

 

[Opalg]

Question: How many elements are in each set?

For the first set, I think it's 8995 because the set is the union of {1,2,3,4,5},{1,2,3,4,5,6},...{1,2,3,...9000}. So 9000 - 5 = 8995.

I think it's 8996, because you need to count both endpoints.

For the second set, I'm not too sure about counting the elements in the set. Since \(\displaystyle 1<x≤i\), I can't think of any x mod i = 2.
For example, I know 5 mod 3 = 2, but 5 > 3 and in this case it wants i to be greater or equal to x...any hints please?

If $i=1$ then the set $\{x\ |\ x \text{ is an integer and } 1<x\leqslant i \text{ and }x=2\pmod i\}$ is the empty set. For all other values of $i$ that set just consists of $x=2$. So your second set is $\emptyset\cup\{2\}$. It therefore contains two elements.

 

[lemonthree]

Both of you are quite right;

For the first question, there are 9000 elements. @Country Boy How do you know that there are 9000 elements though? Doesn't that symbol represent the union of indexed collection from i = 5 to i = 9000? I see it to be similar to the summation notation but I guess that's where I'm wrong.

For the second question, there is 1 element, i.e. {2}, so @Opalg you are right. We take ∅∪{2} to be equal to {2}. Thank you for the explanation, I realized I could view it as 2 = 0*i + 2, for various i values until infinity, which made sense for {2} to be the only element.

 

[HOI]

If $A\subseteq B$ then $A\cup B= B$. These sets are "nested" so the union is just the largest set.

 

[Opalg]

Both of you are quite right;

For the first question, there are 9000 elements. @Country Boy How do you know that there are 9000 elements though? Doesn't that symbol represent the union of indexed collection from i = 5 to i = 9000? I see it to be similar to the summation notation but I guess that's where I'm wrong.

For the second question, there is 1 element, i.e. {2}, so @Opalg you are right. We take ∅∪{2} to be equal to {2}. Thank you for the explanation, I realized I could view it as 2 = 0*i + 2, for various i values until infinity, which made sense for {2} to be the only element.

Yes, you are correct. In both cases I was thinking in terms of a set of sets rather than a union of sets.