Question

Describe without making use of the words "union" and "intersection"

Answer 1

See attachment

Answer 2

the first set represents an open disk : \(x^2+y^2 \lt 1\)

the second set contains a single point : \((0,0)\)

Answer 3

its bit tricky to explain the infinite unions/intersections without using the words union and intersection... you may look at first set like below :


When \(n=1\), the minimum value of radius is \(0\)
As \(n\to \infty\), we have \(\frac{n-1}{n} \to 1\) so clearly the radius sequence has supremum of \(1\)

The infinite collection of all these nested closed disks give an open disk : \(x^2+y^2\lt 1\)

Answer 4

second set is easy to make sense of once you are at peace with first set..

Answer 5

i know right! seemingly impossible haha

Answer 6

hope that animation helps in visulazing why we get an open disk for first infinite set

Answer 7

very nice

Answer 8

i guess im still trying to be at peace witht he first set

Answer 9

the the minimum value is 0 and the max is one i understand this

Answer 10

take ur time :)
key thing to notice is that the radius of circles approach \(1\) as you increase \(n\) but the radius never equals \(1\)

Answer 11

but i dont really understand what he is asking

Answer 12

right right right, it approaches one, sorry

Answer 13

because n-1 / n is tecnically 1 - 1/n

Answer 14

that makes it easy to interpret quickly

Answer 15

The question is simply asking you to figure out the result of that infinite union and explain it with out using the terms "union"

Answer 16

so the sum of the infinite set A and set B would actually equal 1 wouldnt it?

Answer 17

or be less than or equal to one actually

Answer 18

sry i dont get what you're asking/saying :/

Answer 19

http://gyazo.com/b3269040c05458c5a30bc2cd4008caa9

thats the answer for set X

Answer 20

thats ok. so in terms of the explanation, i should just describe what i see, because we are not necessarily looking for an area or a sum of the disks

Answer 21

this is a geometry problem
set X is the collection of all points interior to the circle : x^2 + y^2 = 1

Answer 22

the set of all the interior points of a circle is called open disk : x^2 + y^2 < 1

Answer 23

ok ok im beginning to understand, now, what is set Y intersecting?

Answer 24

Good so are you 100% comfortable with set X ?

Answer 25

yes i understand now, this is an introductory to advanced mathematics class, and i feel like im stuck looking for an actual concrete answer, but i can visualize what set X actually represents now, which im sure is the point of the excersize :)

Answer 26

actual concrete answer for set X is simply : `open disk of radius 1 with center at origin.`

Answer 27

its any circle of the given equation in the range (0 >/ x < 1)

Answer 28

but its radius is not always one is it?

Answer 29

you must know the difference between a `disk` and `circle` to understand these better

Answer 30

let me ask, whats the difference between `disk` and `circle` ?

Answer 31

a disk has an open center.. i hope

Answer 32

look at this picture and try again :)
http://geometry.freehomeworkmathhelp.com/Circles_9/geometry_9_disk_19.gif

Answer 33

hahaha its the opposite >.<

Answer 34

whats the difference ?

Answer 35

sorry to persist lol

Answer 36

please keep persisting i really need help!

Answer 37

the circle is basically the perimeter of the disc

Answer 38

Perfect!

Answer 39

what are the equations of circle and disk of radius 1 ?

Answer 40

(Centered at origin ofcourse)

Answer 41

x^2 + y^2 = 1 for a circle

Answer 42

Right! what about disk

Answer 43

a disk... i dont know

Answer 44

disk is just the region inside circle :

Answer 45

x^2+y^2 < 1 is the equation of OPEN disk of radius 1

x^2 + y^2 <= 1 is the equation of CLOSED disk of radius 1

Answer 46

interresting concept

Answer 47

look at the animation again

Answer 48

the gif?

Answer 49

yes
http://assets.openstudy.com/updates/attachments/54d885bee4b083f72f2a8f1f-ganeshie8-1423479940952-zzz.gif

Answer 50

the obvious difference is that the closed disc includes radius 1

Answer 51

would you agree the union of all those small disks is an OPEN disk of radius 1 ?

Answer 52

as the open disc just infinitely approaches it

Answer 53

open includes all points with the exception of the circle, close inclusedes all points including the circle

Answer 54

you're right

Answer 55

yay :)

Answer 56

let me ask you a question

do you live in USA ?

Answer 57

i do

Answer 58

hawaii!

Answer 59

good, whats the union of hawaii and USA ?

Answer 60

hawaii

Answer 61

wrong, thank mathematically

Answer 62

*think

Answer 63

whats the union of sets {1,2} and {1,2,3,4,5} ?

Answer 64

hmm, hawaii would be a subset, so yeah actually the whole US

Answer 65

Right! another question

whats the union of hawaii, USA and surface of earth ?

Answer 66

the surface of earth, haha

Answer 67

{1,2,3,4,5}

Answer 68

got it!

Answer 69

good, last question :

whats the union of below discs :
x^2 + y^2 <= 0
x^2+y^2 <= 0.1
x^2+y^2 <= 0.2
...
x^2+y^2 <= 0.99999

?