Simple Induction

[daon]

yup! yup!!

edit: Wait... you said is an "intersection instead of union." Just so that you don't come back here and blame me:

\(\displaystyle \bigcap\) - Intersection - What you have above as the answer to [a]

\(\displaystyle \bigcup\) - Union - What you have above as the answer to

 

[dally165]

Yay! Thanks so much! One thing I like bout this forum is the fact that I'm not presented with a direct answer but really help me through!!

Ok I got that part now thanks to you guys!

Here's another one I wana check:

Let A_i = {-i, i} for i ?

with a '+' on the Z's top right saying 'all positive integers' .. i think..

[a] Changes: Union instead of intersection

Changes: none

Answers:

[a] {0} - 1 is not included I think? But 0 will always be!!

infinite number of

Is that correct??

 

[daon]

Try again. I think you are confusing union and intersection.

Unions make "bigger" sets.

\(\displaystyle \{1,2,3\} \cup \{2,3,4\} = \{1,2,3,4\}\)

Intersections make "smaller" sets:

\(\displaystyle \{1,2,3\} \cap \{2,3,4\} = \{2,3\}\).

Also, your answer for above is not even a set.

 

[dally165]

Oh I got it the other way around..

[a] {infinity symbol}

{0}

How's that?

 

[daon]

Oh I got it the other way around..

[a] {infinity symbol}

{0}

How's that?

Sorry, neither are right.

Could you explain why you think the union of all sets \(\displaystyle \{-i,i \}\) is the set \(\displaystyle \{\infty \}\)?

Firstly, in order for something to be in the union or intersection of some sets, it will need to belong to the Universal set... in this case \(\displaystyle \mathbb{Z}^+\).

\(\displaystyle \infty\) is certainly not an integer, and 0 is not a positive one.

 

[dally165]

Hmm.. let me try again:

[a] {Z+}

{1}

How bout that??

 

[daon]

I think you may understand part [a], but it may be my fault you got it wrong from my above post, as I had a typo . \(\displaystyle \mathbb{Z}^+\) was supposed to be \(\displaystyle Z\).

{-1,1}, {-2,2}, {-3,3},...

For the union, [a], we have it looks like every integer (except 0), so the answer is \(\displaystyle \mathbb{Z} \backslash \{0\}\). If took my error into consideration you would have gotten your answer.

In , if 1 were in the intersection then that means 1 belongs to every set {-i,i}, but it definetly isn't in {-2,2} :wink:

 

[dally165]

Haha, ok Im a bit confused now.. What does Z\{0} mean?
And for , I see your point.. it obviously isn't 1. Can I say {Z+}?? coz in a set, there are integers involved?

:?:

 

[mmm4444bot]

What does Z\{0} mean?

In set notation, a backward slash denotes exclusion.

So, you can read the \ character above as "except"

Z \ {0} is the set of all Integers, except 0.

R \ {-1, 0, 1} is the set of all Reals, except -1, 0, and 1.

 

[daon]

Haha, ok Im a bit confused now.. What does Z\{0} mean?
And for , I see your point.. it obviously isn't 1. Can I say {Z+}?? coz in a set, there are integers involved?

:?:

No for . Maybe I'm doing a poor job of explaining it.

Hopefully these examples will aid your intuition:

The intersection of...

{a, b, c}, {a, c}, {a, b} and {1, 2, 3, a, x} = {a}
{ . :shock: , :evil: }, { , :evil: , :mrgreen: , }, and { :idea: , :evil: } = { :evil: }
{-2, -1, 0, 1, 2} and \(\displaystyle \mathbb{Z}^+\) = {1,2}
{2, 4, 6, 8, ...} and {3, 6, 9, 12, ...} = {6, 12, 18, 24, ...}
{1,2,3}, {1,2,3,4} and {4} = {} - the empty set, also referred to as \(\displaystyle \emptyset\) - meaning there are no elements these sets share in common.

What elements do {-1,1} and {-2,2} have in common?

Now what about ?

Also, \(\displaystyle \{\mathbb{Z}^+\}\) and \(\displaystyle \mathbb{Z}^+\) are not the same things. The first is "the set containing the set of positive integers." The second is what you meant to write...

 

[mmm4444bot]

… I don't know why I need to do a Maths paper when i'm majouring in Computers …

Maybe it has something to do with the fact that honing analytical and symbolic reasoning skills (such as those required when studying mathematics) leads individuals to become generally more logical in their thinking, better problem-solvers, and the resultant pruning of the actual connections in the brain to make pathways more efficient doesn't hurt, either.

Maybe it also has something to do with the fact that mathematics is essentially the only way to communicate algorithmic instructions to machines.

Maybe it's to force you to think like a machine. :shock:

 

[dally165]

… I don't know why I need to do a Maths paper when i'm majouring in Computers …

Maybe it has something to do with the fact that honing analytical and symbolic reasoning skills (such as those required when studying mathematics) leads individuals to become generally more logical in their thinking, better problem-solvers, and the resultant pruning of the actual connections in the brain to make pathways more efficient doesn't hurt, either.

Maybe it also has something to do with the fact that mathematics is essentially the only way to communicate algorithmic instructions to machines.

Maybe it's to force you to think like a machine. :shock:

Haha maybe you're right. It makes sense when I'm majouring in programming (software development) but I'm not.. but anyway I guess your right.. it's to train up our minds making it ready and prepared for the future papers/subjects.. lol

Haha, ok Im a bit confused now.. What does Z\{0} mean?
And for , I see your point.. it obviously isn't 1. Can I say {Z+}?? coz in a set, there are integers involved?

:?:

No for . Maybe I'm doing a poor job of explaining it.

Hopefully these examples will aid your intuition:

The intersection of...

{a, b, c}, {a, c}, {a, b} and {1, 2, 3, a, x} = {a}
{ . :shock: , :evil: }, { , :evil: , :mrgreen: , }, and { :idea: , :evil: } = { :evil: }
{-2, -1, 0, 1, 2} and \(\displaystyle \mathbb{Z}^+\) = {1,2}
{2, 4, 6, 8, ...} and {3, 6, 9, 12, ...} = {6, 12, 18, 24, ...}
{1,2,3}, {1,2,3,4} and {4} = {} - the empty set, also referred to as \(\displaystyle \emptyset\) - meaning there are no elements these sets share in common.

What elements do {-1,1} and {-2,2} have in common?

Now what about ?

Also, \(\displaystyle \{\mathbb{Z}^+\}\) and \(\displaystyle \mathbb{Z}^+\) are not the same things. The first is "the set containing the set of positive integers." The second is what you meant to write...

Haha that is a very cute illustration you have there!

So you said that the [a] is Z\{0}

and for it's {1,2,3,4,5,.....} ??? hahahaha im such a pain