Is the set of numbers that can be written as the product of 6 and an integer closed under subtraction?
A.
Yes, because the difference of any two multiples of 6 is equal to 6.
B.
Yes, because the difference of any two multiples of 6 is also a multiple of 6.
C.
No, and a counterexample is 6 – 6 = 0.
D.
No, and a counterexample is 6 – 18 = –12.

- annie12m

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- annie12m

please help

- anonymous

C and D are out right? since both 0 and -12 are multiples of 6

- jchick

We need to get an understanding of what these number look like.
>Is the set of numbers that can be written as the product of 6 and an integer closed
The integers are {...-3, -2, -1. 0, 1, 2, 3, ...}

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## More answers

- jchick

6 times each of those would be:
{..., -18, -12, -6, 0, 6, 12, 18, ...}

- jchick

any two of those numbers and subtract them, will you get an answer that is also in that set.

- jchick

6-6 = 0

- jchick

the difference of any two multiples of 6 is equal to 6.
12 is a multiple of 6
24 is a multiple of 6.
12 - 24 is not equal to 6. So, throw out option A. It is wrong.

- jchick

6 - 6 = 0 which is a multiple of 6 because 0 times 6 = 0.
So, throw out option C.

- jchick

6 - 18 = -12 but -12 is a multiple of 6 because 6 times -2 = -12.
Throw out option D

- jchick

The answer is B because the difference of any two multiples of 6 is also a multiple of 6.

- annie12m

so its ethier one of the yeses

- annie12m

oh ok thanks

- jchick

Well we eliminated A, C, and D

- jchick

So that only left B

- jchick

No problem!

- annie12m

can you help me with 4 more questions?

- jchick

Yes

- annie12m

ok ill give you one at a time

- jchick

Ok thanks!

- annie12m

Numbers p and q are negative integers.
Which statements are true?
Choose all that are correct.
A.
p + q is a negative integer
B.
p • q is a positive integer
C.
p – q is a negative integer
D.
p/ q is a negative integer

- annie12m

A is correct? B is wrong? C not sure. D is wrong?

- annie12m

jckick you there?

- jchick

let's assign p and q numbers
p=3 and q=5

- jchick

Yes

- jchick

is
3+5 an integer?
3-5?
3*5?
3÷5?

- annie12m

well it said p and q are both going to negative

- jchick

A.
p + q is a positive integer
TRUE! Two positive numbers added = a positive number! Always!
B.
p – q is a negative integer
Maybe not. Ie if P = 10 and q = 1,
p-q = 10-1 = +9. FALSE
C.
p • q is a positive integer
Yes. Two positive numbers multiplied together is still a positive integer
D.
is a positive integer

- annie12m

so a and c are correct?

- jchick

Well they said negative right?

- annie12m

yeah

- annie12m

so lets try putting it p and q negative in all the operations

- jchick

Actually wait.

- jchick

Just compare the two statements if they match then select it.

- annie12m

what you mean?

- jchick

c and d

- annie12m

it goes in this order..
A.
p + q is a negative integer
B.
p • q is a positive integer
C.
p – q is a negative integer
D.
p/ q is a negative integer

- jchick

Well A is positive. so that is false

- annie12m

you sure its not a and c?

- annie12m

a negative plus a negative equals a negative

- annie12m

Numbers p and q are negative integers.

- jchick

Oops you are right I misread it. Sorry you are right the first one is negative.

- annie12m

ok

- jchick

Because if p = -2 + q = -2 you get - four.

- annie12m

ya

- jchick

Ok so how about multiplication?

- annie12m

so is it a and c?

- jchick

Tell me the rule for multiplying two negatives.

- annie12m

multipulcation well a negative times a negative or a positive times a positve you get a positive and a negative time a positive or a positive times a negative you get a negative

- annie12m

oh so b would be it too

- jchick

two negatives make a positive

- jchick

In multiplication.

- annie12m

yeah

- jchick

Ok I didn't understand what you had said.

- jchick

So yes it would be B

- annie12m

i said the same thing you said but i said it for all of them

- annie12m

ok now we see subsraction

- jchick

Wait

- jchick

Subtracting a negative by a negative is a positive.

- jchick

-2--2

- jchick

This equals 0

- annie12m

-6 - (-3)
you keep the first number then switch it to addition then change the sign of the last number so now it is -6 plus 3 and that equals -3

- jchick

Yes correct

- annie12m

its still negative so its correct so c is also correct

- annie12m

now lets try division

- jchick

-8/-4 = +2

- annie12m

thats wrong

- jchick

If the 2 numbers you are multiplying or dividing
have the SAME sign (+,+ or -,-) then the result is POSITIVE

- annie12m

so its a, b, and c

- annie12m

next question

- jchick

Correct!

- annie12m

Which sets of numbers are closed under addition?
Choose all answers that are correct.
A.
{0, 2, 5, 8}
B.
even integers
C.
rational numbers
D.
{0}

- annie12m

do you know what rational numbers are?

- jchick

Ok so if you add any two elements of a set and you ALWAYS get another element in the set, that set is closed under addition. For example
2+2=4
2+4=6
and 4 and 6 are part of the even number set.

- annie12m

so even would be correct

- jchick

let's go through each set
a) no 5 + 2 = 7. 7 is not in the set.
b) yes
c) yes
d) 0+0=0

- annie12m

so b c and d?

- annie12m

but what are rational numbers?

- jchick

B.
even integers
C.
rational numbers
D.
{0}

- jchick

a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, p and q, with the denominator, q, not equal to zero.

- jchick

In easy terms A number that can be made by dividing two integers.

- annie12m

but what does that have to do with addition? so its not correct?

- jchick

What I have posted is the correct answer.

- jchick

Closure:
The result of adding two rational numbers is another rational number.

- annie12m

ok ill do those 3 as the answer then

- jchick

Ok

- jchick

Any more questions?

- annie12m

now next question

- jchick

Ok

- annie12m

oh just 2 more left

- jchick

Ok shoot.

- annie12m

Which equations show that the set of whole numbers is not closed under subtraction?
Choose all answers that are correct.
A.
1 – (–2) = 3
B.
1 – 2 = –1
C.
2 – 0 = 2
D.
2 – 4 = –2

- annie12m

oh and stay close attention to where it says NOT

- jchick

Cave, in order to show closure for a set, you need to start with two elements in the set.
(Keep in mind that negative numbers are NOT in the set of Whole numbers).
So you need to look for which one starts with two whole numbers,
then ends up with something that is not a whole number

- jchick

It's not A

- annie12m

b is correct?

- jchick

option A shows us doing subtraction with a negative number.
Negatives are not in the set of whole numbers.
You need to start with this whole number - whole number =

- annie12m

ok

- jchick

Yes B is correct

- jchick

It's not C.
C does not show us leaving the set of whole numbers.
See how the result is still positive.

- annie12m

yea i see

- jchick

D sounds correct,
you start with two elements in the wholes,
then doing subtraction, you left the set of whole numbers.
And B also?
Yay good job \c:/

- annie12m

so b and d got it

- jchick

Ok so last question!

- annie12m

For which operations is the set {0, 1} closed?
Choose all answers that are correct.
A.
multiplication
B.
division
C.
addition
D.
subtraction

- jchick

If we do this:
0−1
Do we end up with a number which is in the set?

- annie12m

ya

- jchick

Are you sure?

- annie12m

wait no

- annie12m

because it equals -1

- jchick

Good so we can eliminate D right?

- annie12m

ya

- jchick

1/0
How bout division?
Do we end up with an element in the set if we do this?

- annie12m

0 +1 equal 1 and 1+0 equals 1 so addition works?

- annie12m

yep it equals 1

- annie12m

so addition and division works

- jchick

So did you understand the last question cave? :O
1+1
We are not closed under addition

- annie12m

oh right forgot about that you can do the number more than once

- jchick

So does the division work?

- annie12m

division works though

- jchick

The whole numbers are closed under addition. There is no way to add two whole numbers to get a nonwhole number.

- annie12m

and multipulcation because 0 times 0 equals 0 and 1 times 1 equals 1 and 1 times 0 equals 0

- jchick

Wait are you sure?

- jchick

Try again.

- annie12m

yeah im sure its division and multipulcation, am i wrong?

- jchick

No division by zero in the land of math.
So division is also a no no.

- annie12m

ok so only multipulcation then?

- jchick

Yes.

- annie12m

ok let me sumbit the quiz and see what i got

- jchick

Ok

- jchick

What did you get?

- annie12m

i failed the last one was multipulcation and division not just multipulcation, the one before the last one was correct, so was the one before the other one, the one before that one was incorrect it was only a and b not a b and c, and the first one was correct

- jchick

Wait that doesn't make any sense.

- jchick

Oh I see on the last one sorry.

- annie12m

so got 2 wrong :( i didnt put division in it and i added one when it was only suppose to be a and b not c

- jchick

The one before was not wrong I don't understand. You should ask your teacher.

- jchick

Wait I said not C

- annie12m

i will, well thanks anyways! well is there any way i can reward you for your time and help? you did help me understand a lot

- jchick

Well you can select best response.

- jchick

But I am sorry if I confused you I did say not C.

- annie12m

ok i did that and i clicked become a fan

- jchick

Thank you so much!

- annie12m

well i gtg now, bye!

- jchick

Bye!

- jchick

@mathmale please look over my work and make sure because apparently I was wrong.

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