Set - Revision history

Cipe: I think the real-world examples can be together with the abstract ones.

2012-08-11T20:42:31Z

I think the real-world examples can be together with the abstract ones.

GregG: /* Examples */ add a concrete example of operations on sets with an explicit list of elements

2012-08-08T15:04:15Z

Examples: add a concrete example of operations on sets with an explicit list of elements

GregG: /* Examples */ it's better to have examples where definiteness is not an issue, even though old examples may be correct and insightful

2012-08-08T14:57:35Z

Examples: it's better to have examples where definiteness is not an issue, even though old examples may be correct and insightful

GregG: add term before we use it

2012-08-08T02:12:53Z

add term before we use it

GregG: /* Application -- Combinatorics */ spacing

2012-08-08T00:53:07Z

Application -- Combinatorics: spacing

GregG: /* Application -- Combinatorics */ no need for formal logic symbols in a basic math article

2012-08-08T00:52:32Z

Application -- Combinatorics: no need for formal logic symbols in a basic math article

GregG: /* Application -- Combinatorics */ add symbol for clarity

2012-08-08T00:51:43Z

Application -- Combinatorics: add symbol for clarity

GregG: /* Examples */ this is a bad example because it not only is irrelevant to math students, it also begs the question of why mathematical union measures marriage quality

2012-08-08T00:51:12Z

Examples: this is a bad example because it not only is irrelevant to math students, it also begs the question of why mathematical union measures marriage quality

GregG: replace/augment section with better examples

2012-08-08T00:49:29Z

replace/augment section with better examples

Cipe: Undo revision 998885 by Werder (talk)

2012-08-08T00:35:04Z

Undo revision 998885 by Werder (talk)

@@ Line 18: / Line 18: @@
 Let <math>A</math> be the set of all [[prime number]]s, and let <math>B</math> be the set of all [[even number]]s.  What is the intersection, <math>A \cap B</math>?  It is a set containing one element, 2.  Since any even number greater than 2 has two as a proper factor, no even number aside from 2 belongs to <math>A</math>.  However, 2 is a prime number.  Thus, the intersection is <math>\{2\}</math>.
 There is the set of unborn children who were [[abortion|aborted]], about which striking conclusions can be drawn.  Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements.  Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.
-Let <math>C</math> be the set of all positive integers that are multiples of <math>3</math>, and let <math>D</math> be the set of all positive integers that have a remainder of <math>1</math> when divided by <math>3</math>.  The intersection, <math>C \cap D</math>, is the empty set, because no positive integer that is a multiple of <math>3</math> will have remainder <math>1</math> when divided by <math>3</math>.  However, if we take <math>B</math> to be the set of even numbers, as before, then <math>C \cap B</math> is not empty.  ''Exercise: What are the elements of <math>C \cap B</math>?''
+Another striking example is the how traditional marriage provides a greater set than otherwise:  the union of <math>A = \{a, b, c, d\}\,</math> and <math>B = \{a, b, c, e\}\,</math> is merely <math>\{a, b, c, d, e\}\,</math>, while the union of a man, <math>M = \{a, b, c, d\}\,</math> and a woman <math>W = \{e, f, g, h\}\,</math>, is <math>\{a, b, c, d, e, f, g, h\}\,</math>, which is a broader and more diverse set.
 == Application -- Combinatorics==

← Older revision		Revision as of 15:04, August 8, 2012
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	== Examples ==		== Examples ==
		+
		+	Let <math>S</math> be the set <math>\{a, b, d, f, h\}</math>, and let <math>T</math> be the set <math>\{d, e, f\}</math>
		+	*The cardinality of <math>S</math> is <math>5</math>.
		+	*The cardinality of <math>T</math> is <math>3</math>.
		+	*The intersection, <math>S \cap T</math> is the set <math>\{d, f\}</math>. This is because <math>d</math> and <math>f</math> are the only two elements in both <math>S</math> and <math>T</math>.
		+	*The union, <math>S \cup T</math> is the set <math>\{a, b, d, e, f, h\}</math>. These six elements are in the union because they belong to <math>S</math>, to <math>T</math>, or to both <math>S</math> and <math>T</math>.
		+	*The set-theoretic difference <math>S \setminus T</math> (alternative notation: <math>S - T</math>) is the set <math>\{a, b, h\}</math> containing all the elements in <math>S</math> that are '''not''' in <math>T</math>.
		+	*The power set of <math>T</math>, <math>\mathcal{P}(T)</math>, consists of all subsets of <math>T</math> including the empty set and <math>T</math> itself. One can check that <math>\mathcal{P}(T) = \{\emptyset, \{d\}, \{e\}, \{f\}, \{d, e\}, \{d, f\}, \{e, f\}, \{d, e, f\}\}</math>. ''Exercise: How many elements are in <math>\mathcal{P}(S)</math>?''

	Let <math>A</math> be the set of all [[prime number]]s, and let <math>B</math> be the set of all [[even number]]s. What is the intersection, <math>A \cap B</math>? It is a set containing one element, 2. Since any even number greater than 2 has two as a proper factor, no even number aside from 2 belongs to <math>A</math>. However, 2 is a prime number. Thus, the intersection is <math>\{2\}</math>.		Let <math>A</math> be the set of all [[prime number]]s, and let <math>B</math> be the set of all [[even number]]s. What is the intersection, <math>A \cap B</math>? It is a set containing one element, 2. Since any even number greater than 2 has two as a proper factor, no even number aside from 2 belongs to <math>A</math>. However, 2 is a prime number. Thus, the intersection is <math>\{2\}</math>.

@@ Line 13: / Line 13: @@
 There is the set of unborn children who were [[abortion|aborted]], about which striking conclusions can be drawn.  Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements.  Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.
-Define "A" as the set of all false assertions, and "B" as the set of all the Bible passages. Since there are no [[Counterexamples to the Bible]], the intersection between set A and set B is the [[empty set]]. On the other hand, if "C" is set of evolutionary assertions, then the intersection between set A and set C is not empty because it contains [[Counterexamples to Evolution]].
+Let <math>C</math> be the set of all positive integers that are multiples of <math>3</math>, and let <math>D</math> be the set of all positive integers that have a remainder of <math>1</math> when divided by <math>3</math>.  The intersection, <math>C \cap D</math>, is the empty set, because no positive integer that is a multiple of <math>3</math> will have remainder <math>1</math> when divided by <math>3</math>.  However, if we take <math>B</math> to be the set of even numbers, as before, then <math>C \cap B</math> is not empty.  ''Exercise: What are the elements of <math>C \cap B</math>?''
 == Application -- Combinatorics==

← Older revision		Revision as of 02:12, August 8, 2012
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−	A '''set''' is a collection of objects. Sets are the subject of [[set theory]]. Two sets are the same if they contain the same elements, regardless of the order they are in. Sets are written in set notation, such as <math>\{1,2\}</math> indicating the set containing the elements 1 and 2. This is the same as the set <math>\{2,1\}</math> because order does not matter when comparing sets.	+	A '''set''' is a collection of objects called elements. Sets are the subject of [[set theory]]. Two sets are the same if they contain the same elements, regardless of the order they are in. Sets are written in set notation, such as <math>\{1,2\}</math> indicating the set containing the elements 1 and 2. This is the same as the set <math>\{2,1\}</math> because order does not matter when comparing sets.

	Also, repetition of elements is irrelevant, so the set <math>\{1,2,2\}</math> is the same as the set <math>\{1,2\}</math>.		Also, repetition of elements is irrelevant, so the set <math>\{1,2,2\}</math> is the same as the set <math>\{1,2\}</math>.

@@ Line 23: / Line 23: @@
 <br />
-Let <math>S = \{\{x,y\}\ |\ x \in X \textrm{ and } y \in Y\}</math>.  The vertical bar means "such that", and the epsilon-like symbol <math>\in</math> means "is an element of".  So this means that S is the collection of sets<ref>Mathematicians don't like to use the phrase "set of sets", since this can lead to some unpleasant paradoxes.  So they say "collection of sets."  A "collection" is really just a set.</ref> which are pairs x and y such that x is in X and y is in Y.
+Let <math>S = \{\{x,y\}\ |\ x \in X\,\textrm{ and }\,y \in Y\}</math>.  The vertical bar means "such that", and the epsilon-like symbol <math>\in</math> means "is an element of".  So this means that S is the collection of sets<ref>Mathematicians don't like to use the phrase "set of sets", since this can lead to some unpleasant paradoxes.  So they say "collection of sets."  A "collection" is really just a set.</ref> which are pairs x and y such that x is in X and y is in Y.
 <br />
@@ Line 32: / Line 32: @@
 <br />
 Now try a harder example, based on the same two sets.  Let
-:<math>G = \{F\ |\ F \subset (X \cup Y) \textrm{ and } \|F\|=2\}</math>.
+:<math>G = \{F\ |\ F \subset (X \cup Y)\,\textrm{ and }\,\|F\|=2\}</math>.
 Now <math>X \cup Y = \{a,b,c,p,q,r\}</math>, so
 :<math>\|X \cup Y\| = 6</math>.
@@ Line 42: / Line 42: @@
 :<math>{n \choose k} = \frac{n!}{k! (n-k)!}</math>.
-In the case of the set <math>G = \{F\ |\ F \subset \{a,b,c,p,q,r\} \textrm{ and } \|F\|=2\}</math>, the set is
+In the case of the set <math>G = \{F\ |\ F \subset \{a,b,c,p,q,r\}\,\textrm{ and }\,\|F\|=2\}</math>, the set is
 :<math>G = \{\{a,b\}, \{a,c\}, \{a,p\}, \{a,q\}, \{a,r\}, \{b,c\}, \{b,p\}, \{b,q\}, \{b,r\}, \{c,p\}, \{c,q\}, \{c,r\}, \{p,q\}, \{p,r\}, \{q,r\}\}\,</math>.
 Its cardinality is 15.

@@ Line 23: / Line 23: @@
 <br />
-Let <math>S = \{\{x,y\}\ |\ x \in X \and y \in Y\}</math>.  The vertical bar means "such that", and the epsilon-like symbol means "is an element of".  So this means that S is the collection of sets<ref>Mathematicians don't like to use the phrase "set of sets", since this can lead to some unpleasant paradoxes.  So they say "collection of sets."  A "collection" is really just a set.</ref> which are pairs x and y such that x is in X and y is in Y.
+Let <math>S = \{\{x,y\}\ |\ x \in X \and y \in Y\}</math>.  The vertical bar means "such that", and the epsilon-like symbol <math>\in</math> means "is an element of".  So this means that S is the collection of sets<ref>Mathematicians don't like to use the phrase "set of sets", since this can lead to some unpleasant paradoxes.  So they say "collection of sets."  A "collection" is really just a set.</ref> which are pairs x and y such that x is in X and y is in Y.
 <br />

← Older revision		Revision as of 00:51, August 8, 2012
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	There is the set of unborn children who were [[abortion\|aborted]], about which striking conclusions can be drawn. Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements. Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.		There is the set of unborn children who were [[abortion\|aborted]], about which striking conclusions can be drawn. Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements. Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.
−
−	Another striking example is the how traditional marriage provides a greater set than otherwise: the union of <math>A = \{a, b, c, d\}\,</math> and <math>B = \{a, b, c, e\}\,</math> is merely <math>\{a, b, c, d, e\}\,</math>, while the union of a man, <math>M = \{a, b, c, d\}\,</math> and a woman <math>W = \{e, f, g, h\}\,</math>, is <math>\{a, b, c, d, e, f, g, h\}\,</math>, which is a broader and more diverse set.

	Define "A" as the set of all false assertions, and "B" as the set of all the Bible passages. Since there are no [[Counterexamples to the Bible]], the intersection between set A and set B is the [[empty set]]. On the other hand, if "C" is set of evolutionary assertions, then the intersection between set A and set C is not empty because it contains [[Counterexamples to Evolution]].		Define "A" as the set of all false assertions, and "B" as the set of all the Bible passages. Since there are no [[Counterexamples to the Bible]], the intersection between set A and set B is the [[empty set]]. On the other hand, if "C" is set of evolutionary assertions, then the intersection between set A and set C is not empty because it contains [[Counterexamples to Evolution]].

← Older revision		Revision as of 00:49, August 8, 2012
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	== Examples ==		== Examples ==
		+
		+	Let <math>A</math> be the set of all [[prime number]]s, and let <math>B</math> be the set of all [[even number]]s. What is the intersection, <math>A \cap B</math>? It is a set containing one element, 2. Since any even number greater than 2 has two as a proper factor, no even number aside from 2 belongs to <math>A</math>. However, 2 is a prime number. Thus, the intersection is <math>\{2\}</math>.

	There is the set of unborn children who were [[abortion\|aborted]], about which striking conclusions can be drawn. Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements. Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.		There is the set of unborn children who were [[abortion\|aborted]], about which striking conclusions can be drawn. Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements. Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.

@@ Line 9: / Line 9: @@
 == Examples ==
-There is the set of poor children, about which striking conclusions can be drawn.  Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements, if they had access to proper nutrition, health care, and educational institutions.  Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.
+There is the set of unborn children who were [[abortion|aborted]], about which striking conclusions can be drawn.  Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements.  Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.
 Another striking example is the how traditional marriage provides a greater set than otherwise:  the union of <math>A = \{a, b, c, d\}\,</math> and <math>B = \{a, b, c, e\}\,</math> is merely <math>\{a, b, c, d, e\}\,</math>, while the union of a man, <math>M = \{a, b, c, d\}\,</math> and a woman <math>W = \{e, f, g, h\}\,</math>, is <math>\{a, b, c, d, e, f, g, h\}\,</math>, which is a broader and more diverse set.