# Thread: Problems with proofs in set theory.

1. ## Problems with proofs in set theory.

Hello I am attempting to teach myself abstract algebra using FM Hall's "Abstract Algebra". The first section is an introduction to set theory which I get concepotually but when goigng through the excercises at the end of the chapter I got completely thrown by the following,

Proove:

A is contained completely in B <=> for all elements x which are not elements of A are elemets of B

How do I go about prooving this. Thanks for any help with this.

(This place could do with a maths notation font or summit )

2. A is contained completely in B <=> for all elements x which are not elements of A are elemets of B
These are not the same. The left-hand side says that A is a subset of B; the right-hand side says that the complement of A ("all elements x which are not elements of A") is a subset of B.

(This place could do with a maths notation font or summit )
You can use LaTeX between math and /math tags (tags use square brackets here unlike angular brackets in HTML). There is also a $\Sigma$ button on the toolbar above the text area where one types posts; it inserts the math tags.

3. Originally Posted by Bwts
Prove:
A is contained completely in B <=> for all elements x which are not elements of A are elemets of B
That statement is false.
This is true $A \subseteq B \Leftrightarrow B^c \subseteq A^c$
Originally Posted by Bwts
(This place could do with a maths notation font or summit)
As you can see we do have symbols. You just have to know LaTeX.
$$A \subseteq B \Leftrightarrow B^c \subseteq A^c$$

4. OK thanks I will retype the question..

Prove $A \subseteq B \iff \forall x.x \notin A, x \in B$

Forgive me if my notation isnt standard as I am learing from a book. The two why arrow is ment to signify a two way implication.

5. Originally Posted by Bwts
Prove $A \subseteq B \iff \forall x.x \notin A, x \in B$
That is also clearly false.This is true: $A \subseteq B \iff \forall x.x \notin B, x \notin A$

6. Hi Plato at the risk of sounding dumb could you please explain?

7. Originally Posted by Bwts
Hi Plato at the risk of sounding dumb could you please explain?
Example: $A = \left\{ {2,3,5,7} \right\}\,\& \,B = \left\{ {0,1,2,3,4,5,6,7,8,9} \right\}$
Now $A \subseteq B$ but $0\notin A$ but $0\in B$.

It must be true that $x \notin {\rm B} \Rightarrow \quad x \notin {\rm A}$.
Every element in $A$ must also be in $B$.
Every element not in $B$ must also be not in $A$.

This may be a translation problem or maybe a notation problem.

8. I am guessing here that the question is implying that B is the universal set, so in this case would it be true that :

$\forall x.x \notin A, x \in B$ ?

9. Originally Posted by Bwts
I am guessing here that the question is implying that B is the universal set, so in this case would it be true that :

$\forall x.x \notin A, x \in B$ ?
There is absolutely nothing in your first post that would imply that.
Once again you must be complete in the question.
Not doing so has wasted a good deal of time.

10. My apologies Plato but the question was exactly how I put it and it seems to be unproovable if it is as you say. I am merely trying to ascertain whether the book is in error or that I am not understanding the procedure.

Most sorry for wasting your time.

11. Originally Posted by Bwts
My apologies Plato but the question was exactly how I put it and it seems to be unproovable if it is as you say. I am merely trying to ascertain whether the book is in error or that I am not understanding the procedure.

Most sorry for wasting your time.
Write EXACTLY what the books says.

12. Originally Posted by drexel28
write exactly what the books says.
i have

13. Well, if:

$\forall x: x \in B$ your whole statement of $\forall x: x \notin A: x \in B$ is completely trivial.
Go with Plato's correction, that actually DOES say something.

14. OK thankyou all for your time.

Iff (se what I did there ) it is not an inconvieniance I will probably be asking more questions as the book, although well written, does now appear to have issues with the excercises.