set elements: B = {-2k/3 ? Z, k ? R | |k| < 9 }

[bson]

My problem is when given a set

B = {-2k/3 ? Z, k ? R | |k| < 9 }

what is the elements of the set ?

thank you!

bson

 

[stapel]

B = {-2k/3 ? Z, k ? R | |k| < 9 }

what is the elements of the set ?

The elements of the set are the real numbers "k" which open the rules "(-2/3)k is an integer" and "the absolute value of k is less than nine".

Find the numbers which fulfill that. If you aren't sure how to proceed, start plugging numbers (between -9 and 9, obviously) into "(-2/3)k", and see what happens. :wink:

Eliz.

 

[bson]

Hi stapel,

My answer is
B = {-5, -4, -3, -2, -1, 0}

Is that correct?

 

[stapel]

My answer is B = {-5, -4, -3, -2, -1, 0}

Is that correct?

How did you arrive at your solution? How did you get that (-2/3)(-5) is an integer, but (-2/3)(-6) was not? :shock:

Please reply with a complete listing of your work and reasoning. Thank you!

Eliz.

 

[bson]

Thank you stapel !
My work as shown below

|k| < 9,it means k = [0,9[ (since it's absolute value)
plug [0,9[ into (-2/3)k
and I got my answer B = {-5, -4, -3, -2, -1, 0}

 

[stapel]

|k| < 9,it means k = [0,9[ (since it's absolute value)

No, I'm afraid that's not what "absolute value" means. The output of |k| will be between zero and nine, but k itself is not restricted to that interval. :shock:

plug [0,9[ into (-2/3)k

Try plugging all of the possible integral values of k into the expression (-2/3)k. :idea:

and I got my answer B = {-5, -4, -3, -2, -1, 0}

You might want to review the definition of the elements of the set B. Are they really just the values of (-2/3)k, or are they the values of k that fulfill the three listed requirements, of which (-2/3)k was one? :wink:

Eliz.

 

[bson]

could you show the answer?

Thank you Eliz.

 

[stapel]

could you show the answer?

Sure, I could, but if you're stuck still not understanding what's going on, how would copying the answer to this exercise help with the next one? :shock:

Let's try to figure out where you're getting stuck. The set B is given as being all the "elements" "k" which fulfill the following requirements:

. . . . .a) k is a real number
. . . . .b) the absolute value of k is less than nine
. . . . .c) when you multiply k by -2/3, you get an integer

Requirement (a) says that "the number k must be any number", so that doesn't narrow things down much.

But what range of numbers can you get from requirement (b)? Is the number -9/2 within this range? If so, does -9/2 fulfill requirement (c), as well? Is the number -9/5 within this range? If so, does -9/5 fulfill requirement (c), as well?

Please reply showing your work and reasoning, so we can "see" where it is that you're getting stuck (something we can't determine from guesses as to the final answer).

Thank you!

Eliz.

 

[bson]

> But what range of numbers can you get from requirement (b)?
[0, 9[

> Is the number -9/2 within this range?
yes, it is since -9/2 = -4.5 and |-4.5| < 9

> If so, does -9/2 fulfill requirement (c), as well?
yes, it does since (-2(-4.5)/3) is integer 3

>Is the number -9/5 within this range?
yes, it is since -9/5 =-1.8 and |-1.8| < 9

>If so, does -9/5 fulfill requirement (c), as well?
No, it doesn't because (-2(-4.5)/3) = 1.2, which doesn't fullfill requirement (c) as it allows only integers.

My answer is little different this time
the range of |k| < 9 = [0,9[
the range of k = -9 < x < 9
and put [-8.999..., 8.999...] into -2k/3
and got B = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

thank you

 

[stapel]

> But what range of numbers can you get from requirement (b)?
[0, 9[

No. Please study what absolute values are and how absolute values work. The values [0, 9) are only half of the possible values of k for which |k| < 9! :shock:

. . . . .Google results for "absolute value"

> Is the number -9/2 within this range?
yes, it is since -9/2 = -4.5 and |-4.5| < 9

Note that your "yes" here contradicts your range above, since -4.5 is not between zero and nine.

> If so, does -9/2 fulfill requirement (c), as well?
yes, it does since (-2(-4.5)/3) is integer 3

Since -9/2 is a real number, since |-9/2| = 9/2 = 4.5 < 9, and since (-2/3)(-9/2) = 3, which is an integer, shouldn't this value then be included in the set B?

>Is the number -9/5 within this range?
yes, it is since -9/5 =-1.8 and |-1.8| < 9

>If so, does -9/5 fulfill requirement (c), as well?
No, it doesn't because (-2(-4.5)/3) = 1.2, which doesn't fullfill requirement (c) as it allows only integers.

Correct: -9/5 is a real number, and |-9/5| = 9/5 = 1.8 < 9, but (-2/3)(-9/5) = 6/5, which is not an integer. So this value should not be included in the set B.

My answer is little different this time
...put [-8.999..., 8.999...] into -2k/3

I'm sorry, but what do you mean when you say that you "put" the numerical interval "into" the expression...? Please reply showing your work and reasoning.

and got B = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

But didn't you show, above, that k = -9/2 should be in the set B...? Shouldn't that contradiction (with your answer here) indicate that there is a problem somewhere...?

In addition, how did you get that (-2/3)(-5) = 10/3 was an integer? Or that (-2/3)(-4) = 8/3, (-2/3)(-2) = 4/3, (-2/3)(-1) = 2/3, (-2/3)(1) = -2/3, (-2/3)(4) = -8/3, or (-2/3)(5) = 10/3 were integers? How did you arrive at this solution?

A different option might be to try following the rule you were given: k must satisfy (-2/3)k = m for some integer in m; that is, (-2/3)k = m = 0, +/-1, +/-2, +/-3, ...., so k = (-3/2)m = (-3/2)(0), (-3/2)(+/-1), (-3/2)(+/-2), ....

Eliz.