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# How to show the value of |X|? watch

1. Question:
Let . Find and find the number of elements of that are divisible by 5.

Woeful Solution
So is 900000 but I am not sure how to show this. Should I define, for ,

.

And then observe and that for and that this family of subsets is mutually disjoint.

Then .

Applying a similar logic for the second part. There are 20 numbers divisible by 5 in so there are 180000 numbers divisible by 5 in .

I feel this is either overkill or there is much neater way of writing this. Any help is greatly appreciated

I'm fairly new to this uni maths lark so I have no clue what they are expecting.
2. (Original post by Cryptokyo)
Question:
Let . Find and find the number of elements of that are divisible by 5.

Woeful Solution
So is 900000 but I am not sure how to show this. Should I define, for ,

.

And then observe and that for and that this family of subsets is mutually disjoint.

Then .

Applying a similar logic for the second part. There are 20 numbers divisible by 5 in so there are 180000 numbers divisible by 5 in .

I feel this is either overkill or there is much neater way of writing this. Any help is greatly appreciated

I'm fairly new to this uni maths lark so I have no clue what they are expecting.
This is way too overcomplicated. How many integers are there between 10 and 20, including 10 and excluding 20? 20-10. The number of integers between a and b (both integers), including a and excluding b is b-a.

So |X| = 1000000 - 100000
3. (Original post by Zacken)
This is way too overcomplicated. How many integers are there between 10 and 20, including 10 and excluding 20? 20-10. The number of integers between a and b (both integers), including a and excluding b is b-a.

So |X| = 1000000 - 100000
Ahh that makes much more sense. I think I might of got a bit carried away .

So would you reason for the number of members of X divisible by 5 by doing 900000/5=180000? Or does it require a bit more justification?
4. (Original post by Cryptokyo)
Ahh that makes much more sense. I think I might of got a bit carried away .

So would you reason for the number of members of X divisible by 5 by doing 900000/5=180000? Or does it require a bit more justification?
I think that's acceptable, but there's a fairly big but... here.

The most common mistake people make with questions like this are "off by one errors". The simplest example: "how many integers between 1 and 10 inclusive? The answer is not 10-1 = 9...".

Put like that, it seems obvious, but similar issues often mess people up, particularly in slightly more complicated problems.

If instead X was the set of integers between 100000 and 1000000 inclusive, then |X| = 900001, and the number of multiples of 5 in X would be 180001.
Or, if X was the set s.t. 100000<=x<100003, then although you can still say |X| = 100003 - 100000 = 3(because we're not including both ends), but you can't say the number of multiples of 5 in X is (100003-100000)/5 = 0.6.

Now, for the particular question you've asked, the off-by-one considerations don't occur, and the simplest method gives the correct results. I think it would be fairly harsh for an examiner to dock marks for you not showing you've considered if there's anything you need to take care of at the end points, but in the interests of getting the right answers, you certainly should consider what happens at the ends.

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