Set Theory Homework find ∪i=0Ai and ∩ i=0Ai

[Uiiop]

Homework Statement

Find ∪i=0Ai (with infinite symbol) and ∩ i=0Ai (with infinite symbol) in each of the cases when for each natural number
i, Ai is defined as:

1. Ai = {i,−i, i + 1,−(i + 1), i + 2,−(i + 2), . . .}
2. Ai = {0, i, 2i}
3. Ai = {x : x is a real number such that i < x < i + 1}
4. Ai = {x : x is a real number strictly between − i and i}.

Homework Equations

The Attempt at a Solution

I really have no clue on this part of set theory

 

[gb7nash]

Homework Statement

Find ∪i=0Ai (with infinite symbol) and ∩ i=0Ai (with infinite symbol) in each of the cases when for each natural number
i, Ai is defined as:

1. Ai = {i,−i, i + 1,−(i + 1), i + 2,−(i + 2), . . .}
2. Ai = {0, i, 2i}
3. Ai = {x : x is a real number such that i < x < i + 1}
4. Ai = {x : x is a real number strictly between − i and i}.

Homework Equations

The Attempt at a Solution

I really have no clue on this part of set theory

What it is that you're confused about? Do you know how to take a union of two sets? What about three sets? ...

In any case, I'm assuming you need to find $\bigcup_{i=0}^{\infty} A_{i}$ for each problem. For these types of problems, just list out A₀, A₁, ...

Let's look at 1.

A₀ = {0,1,-1,2,-2,...} (What is this set?)
A₁ = {1,-1,2,-2,...}
.
.
.

What happens if you take the union of these?

 

[SammyS]

We help with homework. We don't do it for you.

What do you know about any of this?

What part of this problem do you need help with?

For each part of the problem, ( 1., 2., 3., & 4.) write out the first few A_i. In other words, write out A₀, A₁, A₂, A₃, A₄, A₅.

Check you textbook & notes for what is meant by: $\displaystyle \bigcup_{1=0}^\infty \,A_i\,,\ \text{ and }\ \bigcap_{1=0}^\infty \,A_i\,.$

Edited to include A₀. DUH!

 

[Uiiop]

...ok i need to find $\bigcup_{i=0}^{\infty} A_{i}$ & the $\bigcap_{i=0}^{\infty} A_{i}$ for each

i can see that A0 is a set of integers

would the union be?

A0 $\bigcup_$ A1 = {0,1,-1,2,-2,...}

and the intersection?

A0 $\bigcap_$ A1 = {1,-1,2,-2,...}

is that it for question one then? lol i don't know if i have even done it

 

[gb7nash]

Fix up your latex. For the union, the set is correct, but I have no idea what notation you're using. For taking the intersection, what elements do all sets have in common? Like Sammy suggested, write out A₀, A₁, A₂, A₃, ...

 

[Uiiop]

ok
1)
A0 {0,1,-1,2,-2,...}
A1 {1,-1,2,-2,...}
A2 {-1,2,-2,...}
A4 {2,-2,...}
A5 {-2,...}
think that right probably not...

Would $\bigcup_{i=0}^{\infty} A_{i}$ simply be {0,1,-1,2,-2,...} ?
would $\bigcup_{i=0}^{\infty} A_{i}$ be {-2,...}?

 

[gb7nash]

A0 and A1 are correct. A2 is wrong. Look again at the problem and tell me what A2 is. What is A3? A4? etc..

 

[Uiiop]

A2 {0,1,-1,2,-2,3,...}
a3 {1,-1,2,-2,3...}
a4 {0,1,-1,2,-2,3,-3,...}
a5 {1,-1,2,-2,3,-3,...}
?

 

[gb7nash]

Do you understand index notation? A₂ means you plug in 2 wherever i is. For instance, A₁ = {1,-1,1+1,-(1+1),...} = {1,-1,2,-2,...} Knowing this now, what's A₂? A₃? etc..

 

[Uiiop]

as i said clueless ;D
A2 {2,-2,3,-3,4,-4,...}
A3 {3,-3,4,-4,5,-5,...}
A4 {4,-4,5,-5,6,-6,...}
A5 {5,-5,6,-6,7,-7,...}

 

[gb7nash]

as i said clueless ;D
A2 {2,-2,3,-3,4,-4,...}
A3 {3,-3,4,-4,5,-5,...}
A4 {4,-4,5,-5,6,-6,...}
A5 {5,-5,6,-6,7,-7,...}

Those are correct now. What's the union and intersection?

 

[Uiiop]

Would $\bigcup_{i=0}^{\infty} A_{i}$ simply be {0,1,-1,2,-2,3,-3,4,-4,5,-5,6,-6,7,-7,...} ?

would $\bigcap_{i=0}^{\infty} A_{i}$ be {$\o$}?

 

[Uiiop]

for 2) Ai = {0, i, 2i}

A0 {0,0,0}
A1 {0,1,2}
A2 {0,2,4}
A3 {0,3,6}
A4 {0,4,8}
A5 {0,5,10}

Would $\bigcup_{i=0}^{\infty} A_{i}$ simply be {0,1,2,3,4,5,6,8,10}

would $\bigcap_{i=0}^{\infty} A_{i}$ be {0}?

 

[SammyS]

For #1.

Would $\bigcup_{i=0}^{\infty} A_{i}$ simply be {0,1,-1,2,-2,3,-3,4,-4,5,-5,6,-6,7,-7,...} ?

Yes.

would $\bigcap_{i=0}^{\infty} A_{i}$ be {$\o$}?

If by {$\o$} you mean the empty set, then yes.

Empty set = {} = $\emptyset$ .

 

[SammyS]

for 2) Ai = {0, i, 2i}

A0 {0,0,0}
A1 {0,1,2}
A2 {0,2,4}
A3 {0,3,6}
A4 {0,4,8}
A5 {0,5,10}

Would $\bigcup_{i=0}^{\infty} A_{i}$ simply be {0,1,2,3,4,5,6,8,10}

You forgot the "..." .

would $\bigcap_{i=0}^{\infty} A_{i}$ be {0}?

Yes.

 

[Uiiop]

thanks guys i should be ok now no doubt ill be back for help on something else

 

[Uiiop]

...ok now I am stuck on 3) {x : x is a real number such that i < x < i + 1}
how do you list the x?
do you have to declare it?
or do you just leave it as x i think I am barking up the wrong tree but this is what i have got thus far...

a0 = {0,x,1,...}
a1 = {1,x,2,...}
a2 = {2,x,3,...}
a3 = {3,x,4,...}
a4 = {4,x,5,...}
a5 = {5,x,6,...} as i said i don't know what to do with the 'x'

i get that its real numbers and i may have use a different technique problem is i don't know this technique

 

[SammyS]

Do you know about open intervals on the number line ?

If 0 < x < 1 , we write that as (0, 1) , etc.

 

[Uiiop]

no i do not...

so would a0 then be {(0, 1),...} etc
then would the union be
{(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),...} or {0,1,2,3,4,5,6,...}
i imagine the intersection would be $\emptyset$ whichever way

 

[SammyS]

no i do not...

so would a0 then be {(0, 1),...} etc
then would the union be
{(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),...} or {0,1,2,3,4,5,6,...}

Actually, we write this as $(0,1)\cup(1,2)\cup(2,3)\cup(3,4)\cup(4,5)\cup(5,6)\dots$ An alternative way to write this is, { x | x > 0 and x ≠ 1, 2, 3, 4, ...}.
In fact, none of the integers, {0,1,2,3,4,5,6,...} are in the sets A₀ , A₁ , A₁ , ... nor are they in the union of the A_i .

i imagine the intersection would be $\emptyset$ whichever way

Yes, you're right here.

 

[Uiiop]

for 4) {x : x is a real number strictly between − i and i}.

would it be:

a0 {0}
a1 {-1,1}
a2 {-2,2}
a3 {-3,3}
a4 {-4,4}
a5 {-5,5}

union: {0,-1,1,-2,2,-3,3,-4,4,-5,5,...}
intersection: empty set

correct?

 

[Uiiop]

or is it:
{(0)union(-1,1)union(-2,2)union(-3,3)union(-4,4)union(-5,5)...}

 

[SammyS]

or is it:
{(0)union(-1,1)union(-2,2)union(-3,3)union(-4,4)union(-5,5)...}

It's this answer (sort of).

Write (0) as {0}; it's a set. Also each (-1, 1) , (-2, 2), etc., is itself a set .

In fact, (-2, 2) = { x , a real number | -2 < x < 2 }

Here's a graph: [PLAIN]http://www.wolframalpha.com/Calculate/downloadGIF.jsp?podtitle=Number%20line:&podtopright=&podbottom=%7B%20%7D&subpodcount=1&imgloc=/Calculate/MSP/MSP148519g90fga5a3ig3cd000058fi1a99782i1hfg%3FMSPStoreType%3Dimage/gif%26s%3D30%26w%3D310%26h%3D36&rawsubpods=&i=%28-2%2C+2%29&server=http%3A//www4b.wolframalpha.com