Subsets - Revision history

Scott Roby1 at 19:44, 29 June 2015

2015-06-29T19:44:08Z

Scott Roby1: /* Writing Proofs */

2015-06-29T19:36:59Z

Writing Proofs

Scott Roby1 at 19:32, 29 June 2015

2015-06-29T19:32:16Z

Scott Roby1: /* Definition */

2015-06-28T03:01:22Z

Definition

Scott Roby1 at 02:27, 28 June 2015

2015-06-28T02:27:02Z

Scott Roby1 at 02:26, 28 June 2015

2015-06-28T02:26:19Z

Scott Roby1: Created page with " == Definition == Let $X$ and $Y$ be sets. We say that '' $X$ is a subset of $Y$ '' if every element of $X$ is also an e..."

2015-06-28T02:25:31Z

Created page with " == Definition == Let <math>X</math> and <math>Y</math> be sets. We say that ''<math>X</math> is a subset of <math>Y</math>'' if every element of <math>X</math> is also an e..."

New page

== Definition ==
Let <math>X</math> and <math>Y</math> be sets. We say that ''<math>X</math> is a subset of <math>Y</math>'' if every element of <math>X</math> is also an element of <math>Y</math> , and we write <math>X\subseteq Y</math> or <math>Y\supseteq X</math> . Symbolically, <math>X\subseteq Y</math> means <math>x\in X</math> <math>\Longrightarrow</math> <math>x\in Y</math> .

Two sets <math>X</math> and <math>Y</math> are said to be equal, <math>X=Y</math>, if both <math>X\subseteq Y</math> and <math>Y\subseteq X</math>. Note that some authors use the symbol <math>\subset</math> in place of the symbol <math>\subseteq</math>.

← Older revision		Revision as of 19:44, 29 June 2015
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	== Definition ==		== Definition ==
		+	[[File: Subset.png\|thumb\|An example of a set <math>Y</math> and its subset <math>X</math>]]
	Let <math>X</math> and <math>Y</math> be sets. We say that '''''<math>X</math> is a subset of <math>Y</math>''''' if every element of <math>X</math> is also an element of <math>Y</math> , and we write <math>X\subseteq Y</math> or <math>Y\supseteq X</math> . Symbolically, <math>X\subseteq Y</math> means <math>x\in X</math> <math>\Rightarrow</math> <math>x\in Y</math> .		Let <math>X</math> and <math>Y</math> be sets. We say that '''''<math>X</math> is a subset of <math>Y</math>''''' if every element of <math>X</math> is also an element of <math>Y</math> , and we write <math>X\subseteq Y</math> or <math>Y\supseteq X</math> . Symbolically, <math>X\subseteq Y</math> means <math>x\in X</math> <math>\Rightarrow</math> <math>x\in Y</math> .

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 '''How to write a proof that <math>X\subseteq Y</math>''': In general, to show <math>X\subseteq Y</math> we wish to show that if <math>x\in X</math>, then <math>x\in Y</math>. This is done in the following format:
-Let <math>x\in X</math>. ''(logical argument)'', thus <math>x\in Y</math>. This shows that <math>X\subseteq Y</math>.
+'''Proof''' Let <math>x\in X</math>. ''(logical argument)'', thus <math>x\in Y</math>. This shows that <math>X\subseteq Y</math>.
 The logical argument portion often begins by giving the definition of <math>x\in X</math> and ends with the definition of <math>x\in Y</math>.
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 The following is a write-up of the solution of Example 1 as a formal proof:
-Let <math>x\in X</math>. That is, there exists some <math>k\in\mathbb{Z}</math> such that <math>x=6k</math>. We have <math>x=6k=2(3k)</math>. Since <math>3k\in\mathbb{Z}</math>, we also have that <math>x=2n</math> for an integer <math>n</math>, thus <math>x\in Y</math>. This shows that <math>X\subseteq Y</math>.
+'''Proof''' Let <math>x\in X</math>. That is, there exists some <math>k\in\mathbb{Z}</math> such that <math>x=6k</math>. We have <math>x=6k=2(3k)</math>. Since <math>3k\in\mathbb{Z}</math>, we also have that <math>x=2n</math> for an integer <math>n</math>, thus <math>x\in Y</math>. This shows that <math>X\subseteq Y</math>.

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 == Definition ==
 Let <math>X</math> and <math>Y</math>  be sets. We say that '''''<math>X</math> is a subset of <math>Y</math>''''' if every element of <math>X</math>  is also an element of <math>Y</math> , and we write <math>X\subseteq Y</math>  or <math>Y\supseteq X</math> . Symbolically, <math>X\subseteq Y</math>  means <math>x\in X</math>  <math>\Rightarrow</math> <math>x\in Y</math> .
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 We want to show that for any <math>x\in X</math> we also have <math>x\in Y</math>. To do this we will let <math>x</math> be an arbitrary element of the set <math>X</math>. This means that <math>x</math> can be written as <math>x=6k</math> for some integer <math>k</math>. Now we wish to show that <math>x=6k</math> is an element of the set <math>Y</math>. To do this, we need to show that our <math>x</math> satisfies the definition of being an element of <math>Y</math>; that is, <math>x</math> must look like <math>2n</math> for some integer <math>n</math>. This can be seen by writing <math>x=6k=2(3k)=2n</math> and declaring <math>3k=n</math>.

← Older revision		Revision as of 03:01, 28 June 2015
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	== Definition ==		== Definition ==
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	Two sets <math>X</math> and <math>Y</math> are said to be equal, <math>X=Y</math>, if both <math>X\subseteq Y</math> and <math>Y\subseteq X</math>. Note that some authors use the symbol <math>\subset</math> in place of the symbol <math>\subseteq</math>.		Two sets <math>X</math> and <math>Y</math> are said to be equal, <math>X=Y</math>, if both <math>X\subseteq Y</math> and <math>Y\subseteq X</math>. Note that some authors use the symbol <math>\subset</math> in place of the symbol <math>\subseteq</math>.
		+
		+	== Example ==
		+
		+	Show that the set <math>X=\lbrace 6k : k\in\mathbb{Z} \rbrace</math> is a subset of <math>Y=\lbrace 2n : n\in\mathbb{Z} \rbrace</math>
		+
		+	=== Solution ===
		+
		+	We want to show that for any <math>x\in X</math> we also have <math>x\in Y</math>. To do this we will let <math>x</math> be an arbitrary element of the set <math>X</math>. This means that <math>x</math> can be written as <math>x=6k</math> for some integer <math>k</math>. Now we wish to show that <math>x=6k</math> is an element of the set <math>Y</math>. To do this, we need to show that our <math>x</math> satisfies the definition of being an element of <math>Y</math>; that is, <math>x</math> must look like <math>2n</math> for some integer <math>n</math>. This can be seen by writing <math>x=6k=2(3k)=2n</math> and declaring <math>3k=n</math>.

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 == Definition ==
-Let <math>X</math> and <math>Y</math>  be sets. We say that ''<math>X</math> is a subset of <math>Y</math>'' if every element of <math>X</math>  is also an element of <math>Y</math> , and we write <math>X\subseteq Y</math>  or <math>Y\supseteq X</math> . Symbolically, <math>X\subseteq Y</math>  means <math>x\in X</math>  <math>\Rightarrow</math> <math>x\in Y</math> .
+Let <math>X</math> and <math>Y</math>  be sets. We say that '''''<math>X</math> is a subset of <math>Y</math>''''' if every element of <math>X</math>  is also an element of <math>Y</math> , and we write <math>X\subseteq Y</math>  or <math>Y\supseteq X</math> . Symbolically, <math>X\subseteq Y</math>  means <math>x\in X</math>  <math>\Rightarrow</math> <math>x\in Y</math> .
 Two sets <math>X</math> and <math>Y</math> are said to be equal, <math>X=Y</math>, if both <math>X\subseteq Y</math> and <math>Y\subseteq X</math>. Note that some authors use the symbol <math>\subset</math> in place of the symbol <math>\subseteq</math>.