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Math Help - Summation Notation and Math Relationship help

  1. #1
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    Summation Notation and Math Relationship help

    New to the forum and have a maths assignment where I'm struggling to do a few of the questions, most notably the maths relationship questions and summation notation. Any help, steps or advice including reading material would be appreciated.

    Questions struggling with:

    Summation Notation and Math Relationship help-untitled.png

    Doing this course after not doing this math in High School and slowly trying to pick up on it, from my understanding my min is 1/2 and max is 3/4 and it's going up by n over (N+1!) so they're just wanting me to draw the sigma symbol and put those three numbers in?

    Likewise second one?



    I believe I've gotten the first three, but I'm struggling to get my head around understanding questions like number 5,6,7.

    Edit: Thought I'd edit it what I've tried to do. I'm tried re-writing it out so for example number one was Z+ (positive integers) belong to subset of Q. Are all integers rational, yes. Etc. My simplying only seemed to work for first three than I struggled to get my head around the others.

    Summation Notation and Math Relationship help-relatinships.png


    Summation Notation and Math Relationship help-prove.png

    Follows on from struggling to understand the relationship.

    Thanks any help is much appreciated.
    Last edited by Whatthe; May 8th 2014 at 04:17 PM. Reason: Added in extra
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  2. #2
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    Re: Summation Notation and Math Relationship help

    Summation #1: $\displaystyle\sum_{k=1}^n\frac{k}{(k+1)!}$. Try the second one.

    5. $\Bbb Q\cap \Bbb R=\Bbb Q$. The symbol $\cap$ means intersection. For any two sets $A$ and $B$, the set $A\cap B$ contains elements that belong to both $A$ and $B$. So, $\Bbb Q\cap \Bbb R$ contains numbers that are both in $\Bbb Q$ and in $\Bbb R$. But if a number is in $\Bbb Q$, it's also in $\Bbb R$, so $\Bbb Q\cap \Bbb R$ contains the same elements as simply $\Bbb Q$. The statement is true.

    6. $\Bbb Q\cup\Bbb Z=\Bbb Q$. The symbol $\cup$ means union. For any two sets $A$ and $B$, the set $A\cup B$ contains elements that belong to $A$, to $B$ or both. But $\Bbb Z$ is a subset of $\Bbb Q$, so adding it to $\Bbb Q$ does not make the latter set larger. The statement is true.

    Try questions 7 and 8.
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  3. #3
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    Re: Summation Notation and Math Relationship help

    Quote Originally Posted by emakarov View Post
    Summation #1: $\displaystyle\sum_{k=1}^n\frac{k}{(k+1)!}$. Try the second one.

    5. $\Bbb Q\cap \Bbb R=\Bbb Q$. The symbol $\cap$ means intersection. For any two sets $A$ and $B$, the set $A\cap B$ contains elements that belong to both $A$ and $B$. So, $\Bbb Q\cap \Bbb R$ contains numbers that are both in $\Bbb Q$ and in $\Bbb R$. But if a number is in $\Bbb Q$, it's also in $\Bbb R$, so $\Bbb Q\cap \Bbb R$ contains the same elements as simply $\Bbb Q$. The statement is true.

    6. $\Bbb Q\cup\Bbb Z=\Bbb Q$. The symbol $\cup$ means union. For any two sets $A$ and $B$, the set $A\cup B$ contains elements that belong to $A$, to $B$ or both. But $\Bbb Z$ is a subset of $\Bbb Q$, so adding it to $\Bbb Q$ does not make the latter set larger. The statement is true.

    Try questions 7 and 8.
    Thank you very much for explaining both five and six I think I understand it now.

    For 7 Z+ intersection R= Z+ I believe that's true as theirs both rational numbers and integers in Q.
    For 8 I also got true for similar reasons. I think they're all true except 3?

    In regards to summation notation question still having a bit of trouble with second one;

    I see how you did the first one, so with the second one:

    The denominator is obviously increasing by one, and the numerator is subtracting by one more every time, and they're both the same number. Certainly need to revise summation notation.

    thanks
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  4. #4
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    Re: Summation Notation and Math Relationship help

    Quote Originally Posted by Whatthe View Post
    For 7 Z+ intersection R= Z+ I believe that's true as theirs both rational numbers and integers in Q.
    Question 7 does not mention rational numbers or $\Bbb Q$.

    Quote Originally Posted by Whatthe View Post
    For 8 I also got true for similar reasons.
    Imagine that $\Bbb Q$ is a set of fruits (apples, oranges, pears, etc.) and $\Bbb Z$ is the set of apples in $\Bbb Q$. That is, $\Bbb Z$ is a subset of $\Bbb Q$. If you take the union of apples and all fruit, do you get just apples?

    There is nothing tricky about these questions. The principle behind them is clear to kindergarten students. The only reason I imagine people can get them wrong is when they don't know the definitions of sets or operations, mistake, say, $\cup$ for $\cap$ or are intimidated by the notation. Once you feel confident about the concepts involved, I can't see how you can get them wrong.

    The wrong statements are 2, 3 and 8. Whether 4 is true depends on the definition of $\Bbb Z^+$ and $\Bbb Z^-$. If neither of them includes 0, then statement 4 is false. Forgetting about 0 (e.g., saying "positive" when the correct term is "nonnegative") is a common error.

    Quote Originally Posted by Whatthe View Post
    The denominator is obviously increasing by one, and the numerator is subtracting by one more every time, and they're both the same number.
    The relative change between terms ("the denominator is increasing by one") is irrelevant. What you need to come up with is the general form of each term separately. The $k$'th denominator is $k!$. Numerator #1 is $n$. Numerator #2 is $n-1$. Numerator #3 is $n-2$. Numerator #4 is $n-3$. Numerator #$(n-1)$ is 2. Finally, numerator #$n$ is 1. Can you guess the relationship between a number $k$ and the numerator #$k$?
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