Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    14
    ReputationRep:
    Question:
    Let X=\left\lbrace x\in\mathbb{Z}:100000\leq x < 1000000\right\rbrace. Find |X| and find the number of elements of X that are divisible by 5.

    Woeful Solution
    So |X| is 900000 but I am not sure how to show this. Should I define, for i\in\mathbb{Z}:1\leq i\leq 9000,

    A_{i}=\left\lbrace a\in X: 99900+100i \leq x <100000+100i \right\rbrace.

    And then observe A_{i}\subset X and that |A_{k}|=|A_{l}| for k,l\in\mathbb{Z}:1\leq k,l\leq 9000 and that this family of subsets is mutually disjoint.

    Then |X|=9000|A_{i}|=9000\times100=90  0000.

    Applying a similar logic for the second part. There are 20 numbers divisible by 5 in A_{1} so there are 180000 numbers divisible by 5 in X.

    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.
    Offline

    22
    ReputationRep:
    (Original post by Cryptokyo)
    Question:
    Let X=\left\lbrace x\in\mathbb{Z}:100000\leq x < 1000000\right\rbrace. Find |X| and find the number of elements of X that are divisible by 5.

    Woeful Solution
    So |X| is 900000 but I am not sure how to show this. Should I define, for i\in\mathbb{Z}:1\leq i\leq 9000,

    A_{i}=\left\lbrace a\in X: 99900+100i \leq x <100000+100i \right\rbrace.

    And then observe A_{i}\subset X and that |A_{k}|=|A_{l}| for k,l\in\mathbb{Z}:1\leq k,l\leq 9000 and that this family of subsets is mutually disjoint.

    Then |X|=9000|A_{i}|=9000\times100=90  0000.

    Applying a similar logic for the second part. There are 20 numbers divisible by 5 in A_{1} so there are 180000 numbers divisible by 5 in X.

    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
    • Thread Starter
    Offline

    14
    ReputationRep:
    (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 :crazy:.

    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?
    Offline

    17
    ReputationRep:
    (Original post by Cryptokyo)
    Ahh that makes much more sense. I think I might of got a bit carried away :crazy:.

    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.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    What newspaper do you read/prefer?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.