Difference between revisions of "Subsets"
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== Definition == | == 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> . | ||
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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>. | 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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| + | == Writing Proofs == | ||
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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: | ||
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| + | Let <math>x\in X</math>. ''(logical argument)'', thus <math>x\in Y</math>. This shows that <math>X\subseteq Y</math>. | ||
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| + | 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: | ||
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| + | 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>. | ||
Revision as of 11:32, 29 June 2015
Definition
Let and be sets. We say that is a subset of if every element of is also an element of , and we write or . Symbolically, means .
Two sets and are said to be equal, , if both and . Note that some authors use the symbol in place of the symbol .
Example
Show that the set is a subset of
Solution
We want to show that for any we also have . To do this we will let be an arbitrary element of the set . This means that can be written as for some integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} . Now we wish to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=6k} is an element of the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} . To do this, we need to show that our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} satisfies the definition of being an element of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} ; that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} must look like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} for some integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . This can be seen by writing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=6k=2(3k)=2n} and declaring Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3k=n} .
Writing Proofs
How to write a proof that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\subseteq Y} : In general, to show Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\subseteq Y} we wish to show that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in Y} . This is done in the following format:
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} . (logical argument), thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in Y} . This shows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\subseteq Y} .
The logical argument portion often begins by giving the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} and ends with the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in Y} .
The following is a write-up of the solution of Example 1 as a formal proof:
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in X} . That is, there exists some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\in\mathbb{Z}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=6k} . We have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=6k=2(3k)} . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3k\in\mathbb{Z}} , we also have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2n} for an integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in Y} . This shows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\subseteq Y} .