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#1 2014-03-26 04:12:41

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

interval in sets

I have just started studying sets.
I couldn't understand intervals as subset of R, The set of real numbers

there are 4 kinds of intervals given:
1.open interval
2.closed interval
3.right half open interval
4.left half open interval

pls. explain it clearly.


friendship is tan 90°.

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#2 2014-03-26 04:17:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: interval in sets

What we use in interval arithmetic:

The interval of numbers between a and b, including a and b, is often denoted [a, b]. The two numbers are called the endpoints of the interval.

The interval of numbers between a and b, not including a and b, is often denoted (a, b).

( means open, ] means closed.

An open interval does not include its endpoints, and is indicated with parentheses. For example (0,1) means greater than 0 and less than 1. A closed interval includes its endpoints, and is denoted with square brackets. For example [0,1] means greater than or equal to 0 and less than or equal to 1.

(1,2,3,4,5,6,7,8,9,10 ) does not include 1 or 10.

[ 1,2,3,4,5,6,7,8,9,10] does include both 1 and 10.

[ 1,2,3,4,5,6,7,8,9,10) Includes 1 and not 10.

( 1,2,3,4,5,6,7,8,9,10 ] includes 10 but not 1.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2014-03-26 05:53:15

eigenguy
Member
Registered: 2014-03-18
Posts: 78

Re: interval in sets

Except that these are sets of real numbers, not integers, so you can't list all the members of them (not saying that bobbym is wrong - he is just giving examples of intervals of integers, not intervals of real numbers).

(a, b) = set of all real numbers x such that  a < x and x < b = { x | a < x  and  x < b}
(a, b] = set of all real numbers x such that  a < x and x ≤ b = { x | a < x  and  x ≤ b}
[a, b) = set of all real numbers x such that  a ≤ x and x < b = { x | a ≤ x  and  x < b}
[a, b] = set of all real numbers x such that  a ≤ x and x ≤ b = { x | a ≤ x  and  x ≤ b}

(In all of those, it is assumed that a < b. Some people allow a = b, in which case the first three are empty, and [a, a] = {a}, a singleton set containing just a. Some people also allow b < a, in which case all four sets are empty. Usually, though intervals are only given with a < b.)

So the set (0, 1) contains 0.1, 0.01, 0.001, 0.0001, etc., no matter how many 0s occur before the 1. I also contains 0.9, 0.99, 0.999, etc (for any finite number of 9s), because all of the these numbers are > 0, but < 1. It does not contain either 0 or 1.
The set (0, 1] contains any number in the open interval (0,1), but also contains 1 (but still not 0).
Similarly, the set [0, 1) contains everything on (0,1), but also includes 0, but not 1.
Finally, [0, 1] contains 0, 1, and all the numbers in (0,1).

The next difficulty is that sometimes we want everything less than a number, or everything greater than a number. For these, we replace a with -∞ or b with ∞. But since ∞ is not a real number, it cannot be a part of the set, so it is only given with an open side:
(-∞, b) = the set of all real numbers x such that x < b = { x | x < b}
(-∞, b] = the set of all real numbers x such that x ≤ b = { x | x ≤ b}
(a, ∞) = the set of all real numbers x such that a < x = { x | a < x}
[a, ∞) = the set of all real numbers x such that a ≤ x = { x | a ≤ x}
(-∞, ∞) = the set of all real numbers.


"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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#4 2014-03-26 19:47:46

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

Re: interval in sets

thanks.


friendship is tan 90°.

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