What is the difference between a "proper subset"  and regular "subset" in mathematics?
830686
A subset of a set contains any or all of the elements of the set.\n\nA proper subset contains any of the elements of the set but not all of them.
523097
The ∞ mean infinity.  So, it goes from -infinity to 8, not including 8 itself.  And the other one goes from 5 to infinity, including 5.\n\nThe [ ] means that you include that end point, whereas ( ) means you don't include that point.
942422
You can start from the following identity for set differences which holds true for any sets X, A, and B\n\nX − (A ∪ B)  =  (X − A) ∩ (X − B)\n\nNow, if B is a subset of A, then A ∪ B = A\n\nthe identity above becomes\n\nX − A =  (X − A) ∩ (X − B)\n\nFor this to hold true, X − B must be a superset of (X − A), or in other words, (X − A) must be a subset of X − B
905291
The previous result is that the number of elements in the preimage of each element in f(K) (or more generally in f(G)) is n. All those sets are disjoint and their union is K, so the total number of elements in K is n * (Number of elements in f(K)).\n --> |K| is a multiple of |f(K)|\n\nAnother way of looking at is that if N is the kernel of f, f(K) is isomorphic to the quotient group K/N which always divides the order of K.
2201
Scheme is not a subset of Common Lisp; it is a separate dialect of Lisp that was designed to be much smaller and somewhat more elegant. It is intended primarily for teaching and research although it has many other uses as well.  Scheme is a small language, hence easy to implement, so there are many implementations around.
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There are many different sets, they are typically divided by use.\n\nA few sets are\nReal, Rational, Irrational, Infinitive, Whole, Natural
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The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
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The question was not clear exactly what subsets of line?\nHere are a few possibilities:\n\n1.Subset of a real line http://en.wikipedia.org/wiki/Real_line ?  \n“In mathematics, the real line is simply the set R of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space.” See also real numbers http://en.wikipedia.org/wiki/Real_number#Basic_properties_of_the_real_numbers \nLike in a subset of integers (3, 4, 5, 6) or (3.10, 3.11, 3.12…) \nExtension: \n2.In what geometry?\n    Point in a segment form an infinite subset of a line in Euclidian geometry.\n3.Is a line made up of segments? \n    Then the segments form a subset of line segments then?\n4.Clustering ?\n   Sets of points belonging to a line that cluster around certain conditions become subsets of line.\n\n\nCan you supply more specific information?\n\nI hope that helps.
1163625
Improper fractions are much easier to deal with in any math, physics, or chemistry equation than mixed numbers.\n\nThe only time mixed numbers are easier are for graphing purposes or in explaining your results to someone without a math background (which, sadly, is most everybody that doesn't post here).\n\n;-)
805944
if B' is subset of A' then\n\nif x belongs to B' then x belongs to A'\n\nif x belongs to A then x cannot belong to B' because if x belongs to B' then it belongs to A then x belongs to A and A'\n\nso if  x belongs to A the x belongs to B(as it does not belong to B')\n\nso A is subset of B
245111
As with many things, this depends on what you mean by 'more'. The collection of even integers is a proper subset of the set of integers, so in that sense there are more integers than even integers. \n\nBut when mathematicians talk about this issue, more or less is described by something called 'cardinality'. We say two sets have the same cardinality if there is a way of pairing off the two sets. In this sense, there are the same number of integers as even integers because you can pair every integers with its double.\n\nIt should be pointed out that there are infinite sets that have different cardinalities. The collection of all real numbers is larger than the set of integers even in this sense.
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This is actually the special case of the von Neumann's construction of natural numbers. This is also regarded as the standard construction of natural numbers in set theory.\n\nBefore we construct the set of natural numbers, we must first define succession, which was stated by Peano axioms as follows:\n[1] There is a natural number "0".\n[2] Every natural number a has a natural number successor, denoted by a+. \n[3] There is no natural number whose successor is 0; i.e. no such x exists such that x+ = "0".\n[4] If a ≠ b then a+ ≠ b+. In other words two different natural numbers have different successors. \n[5] If a property is possessed by "0" and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.) \n\nNote that the "0" in the said axiom is just a label and not the usual 0 that we know. In other words, the "0" here is just a starting point in which when successively applied by a successor function (+) will satisfy the Peano's axioms.\n\nNow, to construct the set of natural numbers, we do the stardard way. That is we define 0 to be { } and define the successor element, a+ to be a+ = a ∪ {a} for every set a. (Note: we read a+ to be the successor of a). Then we have the following:\n0 = { }\n1 = {0} = {{ }}\n2 = {0, 1} = {0, {0}} = {{ }, {{ }}}\n3 = {0, 1, 2} = {0, {0, {0, {0}}}} =  {{ }, {{ }}, {{ }, {{ }}}} \nand so on.\n\nTo generalize, we get\nn = {0,1,2,…,n−2,n−1} = {0,1,2,…,n−2} ∪ {n−1} = (n−1) ∪ {n−1}.\n\nFrom this construction, we see that there are exactly n elements set n. So, n ≤ m if and only if n is a subset of the set m.
71038
Looks like the first 4 answerers have a good imagination but know nothing about countable sets\n\nSpecifically:\n\n> if you take the real number system,\nit extends from -infinity to +infinity.\n\nLOL, how does that answer the question?\n\n> I think it's uncountable. Yeah, definitely.\n\nOh, are you sure? I mean, really, really sure?\nHow about a reason?\n\n> Uncountable. because between any two real numbers there are infinite number of real numbers.\n\nSo what? Infinite can be countable or not.\n\n> however if u think they are countable, give it a try\n\nLOL, really?\nLet me see you count all natural numbers then (which ARE countable)\n\n> The set of real numbers is uncountable as it contains all the positive natural numbers and the natural numbers are uncountable.\n\nThe natural numbers ARE countable by definition!!!!!\nDo you even know what countable means?!\n\n\n\nOk, let me give a serious answer now.\nThe set of real numbers is uncountable. It was proven by Cantor, in at least two ways.\nThe proofs are not simple enough to be copied here, so I'll just refer you to them.\n\nHis first argument: http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof\nHis diagonal argument (best known): http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument\n\nRead these pages and follow the proofs.
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Yes! Big difference. Arithmetic has to do with simple operations of only numbers involving +, -, /, and x. (adding, subtracting, dividing, and multiplying). Mathematics encompasses EVERY kind of math, from arithmetic to calculus to linear algebra to the stuff they used for the physics to build the atomic bomb.
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Additive equivalence is a symmetric relation over the reals. See the source link.
877843
do you *really* think you're going to get help in set theory from Yahoo Answers?\n\ntry the Wikipedia Reference Desk (http://en.wikipedia.org/wiki/Wikipedia:Reference_desk ).
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I'm partially guessing because I'm working from memory.\n\nA set is a list of unique values\nA Map is a list of Key/Value pairs.