How Many Elements Are Found in Each Mathematical Set?

In summary: The first set contains 8995 elements because it is the union of sets {1,2,3,4,5}, {1,2,3,4,5,6}, ... , {1,2,3,...,9000}, which each contain 5 elements. The second set contains 1 element, {2}, because it is the union of an empty set and the set {2}. In summary, the first set contains 8995 elements and the second set contains 1 element.
  • #1
lemonthree
51
0
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?
counting-sets.png
 
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  • #2
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.
 
  • #3
lemonthree said:
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.

lemonthree said:
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.
 
  • #4
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.
 
  • #5
If $A\subseteq B$ then $A\cup B= B$. These sets are "nested" so the union is just the largest set.
 
  • #6
lemonthree said:
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.
 

FAQ: How Many Elements Are Found in Each Mathematical Set?

How do you count the elements in a set?

To count the elements in a set, you can simply count the number of distinct objects or values in the set. This can be done by listing out each element and keeping track of the total number as you go.

Can the number of elements in a set change?

Yes, the number of elements in a set can change. If new elements are added to the set, the total number of elements will increase. Similarly, if elements are removed from the set, the total number will decrease.

What is the difference between a finite and infinite set?

A finite set contains a specific and limited number of elements, while an infinite set has an unlimited number of elements. For example, a set of all even numbers is infinite, while a set of all letters in the alphabet is finite.

How do you know if two sets have the same number of elements?

If two sets have the same number of elements, they are said to have the same cardinality. To determine if two sets have the same cardinality, you can count the elements in each set and compare the totals. If they are equal, the sets have the same number of elements.

Can a set have no elements?

Yes, a set can have no elements. This is known as an empty set, or a null set. It is denoted by the symbol ∅ and is different from a set with one element, as it contains no elements at all.

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