1. ## Proving subsets =S

Hello
Iv got an exam in a week and am trying to get through my whole course in that week. And am off to a horrible start =/
Can someone please tell how i should go about proving questions like the follwoing.

Q1)

A = {x <element_of> R | cos x = 1}
A = {x <element_of> R | sin x = 0}
Show that A <subsest of> B

Q2)

X = {24k + 7 | k <element_of> Z}
Y = {4n + 3 | n <element_of> Z}
Z= {6m + 1 | m <element_of> Z}

prove: X <subset_of> Y and X <subset_of> Z but Y <not_subset_of> Z

Thanks and im sure alot more questions would be comming soon =(

2. Originally Posted by lost1
Hello
Iv got an exam in a week and am trying to get through my whole course in that week. And am off to a horrible start =/
Can someone please tell how i should go about proving questions like the follwoing.

Q1)

A = {x <element_of> R | cos x = 1}
A = {x <element_of> R | sin x = 0}
Show that A <subsest of> B

Q2)

X = {24k + 7 | k <element_of> Z}
Y = {4n + 3 | n <element_of> Z}
Z= {6m + 1 | m <element_of> Z}

prove: X <subset_of> Y and X <subset_of> Z but Y <not_subset_of> Z

Thanks and im sure alot more questions would be comming soon =(
Appeal to the definition of subset. $\displaystyle X \subset Y \Leftrightarrow \left(x\in X\Rightarrow x\in Y\right)$

So for the first one. We want to show that if $\displaystyle x\in X\implies\cos(x)=1$ that $\displaystyle x\in Y\implies\sin\left(x\right)=0$.

Can you do that?

3. Hey thanks for the quick reply =)

hmm...yess i tried using the basic idea but couldnt get a proper proof. Would this be a clear enough proof:

Let x <element_of> A, that is cos x = 1
ie. x = 2kPi where k <element_of> N

Now, sin(x) = sin(2kPi) = 0 and therefor x <element_of> B
That is, A <subset_of> B

?

4. also, how do i show the maths symbols nicely like u have?

5. Originally Posted by lost1
Hey thanks for the quick reply =)

hmm...yess i tried using the basic idea but couldnt get a proper proof. Would this be a clear enough proof:

Let x <element_of> A, that is cos x = 1
ie. x = 2kPi where k <element_of> N

Now, sin(x) = sin(2kPi) = 0 and therefor x <element_of> B
That is, A <subset_of> B

?
Did you take a non-decript element of $\displaystyle A$. Yes. Did you show that this element must be an element of $\displaystyle B$. Yes. Did you give a complete and correct proof. Yes.

6. Originally Posted by lost1
also, how do i show the maths symbols nicely like u have?
Enclose them in math brackets For example. to do the $\displaystyle \underbrace{X\cup Y=\int_0^{\Phi}f_n(x)dx}_{\text{this}}$ you must merely enter
Code:
\underbrace{X\cup Y=\int_0^{\Phi}f_n(x)dx}_{\text{this}}
this enclosed in [ math] [/tex].

7. thanks =)
now how do i go about doing the second one?

8. Originally Posted by lost1
thanks =)
now how do i go about doing the second one?
Well $\displaystyle 24k+7=4(6k)+4+3=4(6k+1)+3$ And clearly $\displaystyle 6k+1\in\mathbb{Z}$.......so

9. ohh god i am thick arnt I...

10. Originally Posted by lost1
ohh god i am thick arnt I...
Nah. Little stupid things like that are always the hardest.

11. ok another quick question...

Prove if
$\displaystyle A <intersect> C \subset B <instersect> C$
and
$\displaystyle A \cup C \subset B \cup C$

then $\displaystyle A \subset B$

My start at proof:
Let $\displaystyle x \in A$ therefore $\displaystyle x \in B \cup C$

and then =/

also whats the code for intersection?