Proving that A/B=C: A is set of all integers, B = {x = 2n+5, n an integer}, C = {x = -2m, m an integer}

[tbramer]

The problem gives the following information:

A is an integer ( A = Z )

In the set of B, x is an integer such that x=2n+5 for some n integer ( B = { x = Z : x=2n+5 for some n = Z } )

C = { x = Z : x=-2m for some m = Z }

I need to prove that A/B = C

I know that the problem will need me to prove that C is a subset of A/B and that A/B is a subset of C.

I started by trying to prove that A/B is a subset of C.

My work:

Let A exist as an integer such that B=AC (Too early to use A/B (A divides B) = C in my proof?)

 

[stapel]

A is an integer ( A = Z )

The set A can't be "an" integer, but it can be equal to the set of all integers, often denoted as [imath]\Z[/imath]. Is this what you meant?

In the set of B, x is an integer such that x=2n+5 for some n integer ( B = { x = Z : x=2n+5 for some n = Z } )

Do you mean "in the set B"? (If not, please explain the "of" in the posted expression.) Also, "x = Z" means that x is the set of all integers. Did you perhaps mean that x is "in" Z, so that x is an integer? So maybe the set is actually [imath]B = \{x \in \Z\, |\, x = 2n + 5 \text{ for } n \in \Z\}[/imath]?

C = { x = Z : x=-2m for some m = Z }

Same questions here. Is the set actually [imath]C = \{x \in \Z\, |\, x = -2m \text{ for } m \in \Z\}[/imath]?

I need to prove that A/B = C

Does the notation "A/B" mean "A-complement-B"? If not, what is the meaning?

I know that the problem will need me to prove that C is a subset of A/B and that A/B is a subset of C.

Yes; double set inclusion is the standard way to prove set equality.

I started by trying to prove that A/B is a subset of C.

My work:

Let A exist as an integer such that B=AC (Too early to use A/B (A divides B) = C in my proof?)

What do you mean by the set "AC"? What do you mean by "Let A exist as an integer"? Isn't A a set, equal to the set of integers? What do you mean by one set "dividing" another?

Thank you!

Eliz.

 

[Steven G]

Can you define what you mean when you divide sets?