Grad Wiki - User contributions [en]

Intersections

2015-06-29T20:12:27Z

Scott Roby1: /* Definition */

== Definition ==
[[File:Intersections.png|thumb|The Venn diagram displays two sets <math>X</math> and <math>Y</math> with the intersection <math>X\cap Y</math> shaded]]
Let <math>X</math> and <math>Y</math> be subsets of some universal set <math>U</math>. The '''''intersection of <math>X</math> and <math>Y</math>''''', written <math>X\cap Y</math>, is the set of all <math>x</math> in <math>U</math> which are in both of the sets <math>X</math> and <math>Y</math>.<br />
Symbolically, <math>X\cap Y=\lbrace x\in U : x\in X \text{ and } x\in Y\rbrace</math>.

== Examples ==

=== Example 1 ===
Determine the intersection of the sets <math>X=\lbrace 1,2,5,8 \rbrace</math> and <math>Y=\lbrace 1,3,6,9 \rbrace</math>.

'''Solution.''' By definition, we wish to find the set of all elements which are in both of the sets. The only such element is <math>1</math>. Thus, our solution is <math>X\cap Y=\lbrace 1 \rbrace</math>.

=== Example 2 ===
Prove that for any sets <math>X</math> and <math>Y</math>, <math>X\cap Y\subseteq X</math>.

'''Proof.''' Let <math>x\in X\cap Y</math>. That is, <math>x\in X</math> and <math>x\in Y</math>. In particular, since <math>x\in X</math> we have that <math>X\cap Y\subseteq X</math>.

File:Intersections.png

2015-06-29T20:11:36Z

Scott Roby1:

Intersections

2015-06-29T20:09:55Z

Scott Roby1: Created page with "== Definition == Let <math>X</math> and <math>Y</math> be subsets of some universal set <math>U</math>. The '''''intersection of <math>X</math> and <math>Y</math>''''', writte..."

== Definition ==
Let <math>X</math> and <math>Y</math> be subsets of some universal set <math>U</math>. The '''''intersection of <math>X</math> and <math>Y</math>''''', written <math>X\cap Y</math>, is the set of all <math>x</math> in <math>U</math> which are in both of the sets <math>X</math> and <math>Y</math>.<br />
Symbolically, <math>X\cap Y=\lbrace x\in U : x\in X \text{ and } x\in Y\rbrace</math>.

== Examples ==

=== Example 1 ===
Determine the intersection of the sets <math>X=\lbrace 1,2,5,8 \rbrace</math> and <math>Y=\lbrace 1,3,6,9 \rbrace</math>.

'''Solution.''' By definition, we wish to find the set of all elements which are in both of the sets. The only such element is <math>1</math>. Thus, our solution is <math>X\cap Y=\lbrace 1 \rbrace</math>.

=== Example 2 ===
Prove that for any sets <math>X</math> and <math>Y</math>, <math>X\cap Y\subseteq X</math>.

'''Proof.''' Let <math>x\in X\cap Y</math>. That is, <math>x\in X</math> and <math>x\in Y</math>. In particular, since <math>x\in X</math> we have that <math>X\cap Y\subseteq X</math>.

Math 144

2015-06-29T20:07:05Z

Scott Roby1: /* Pages In Progress */

== General Resources ==

books, blurbs etc

== Pages In Progress ==

[[Truth Tables]]

[[Subsets]]

[[Unions]]

[[Intersections]]

== Pages For Review ==

== Exams ==

== Files ==
any uploaded file relating to set theory goes here

Unions

2015-06-29T20:05:28Z

Scott Roby1: /* Definition */

== Definition ==
[[File:Union.png|thumb|The Venn diagram displays two sets <math>X</math> and <math>Y</math> with the union <math>X\cup Y</math> shaded]]
Let <math>X</math> and <math>Y</math> be subsets of some universal set <math>U</math>. The '''''union of <math>X</math> and <math>Y</math>''''', written <math>X\cup Y</math>, is the set of all <math>x</math> in <math>U</math> which are in at least one of the sets <math>X</math> or <math>Y</math>.<br />
Symbolically, <math>X\cup Y=\lbrace x\in U : x\in X \text{ or } x\in Y\rbrace</math>.

== Examples ==
=== Example 1 ===
Determine the union of the sets <math>X=\lbrace 1,2,5,8 \rbrace</math> and <math>Y=\lbrace 1,3,6,9 \rbrace</math>.

'''Solution''' By definition, we wish to find the set of all elements which are in at least one of the two sets. Thus, we will collect all unique elements into a set as long as they appear once or more between the sets. Our solution is <math>X\cup Y=\lbrace 1,2,3,5,6,8,9 \rbrace</math>.

=== Example 2 ===
Prove that for any sets <math>X</math> and <math>Y</math>, <math>X\subseteq X\cup Y</math>.

'''Proof''' Let <math>x\in X</math>. We wish to show that <math>x\in X\cup Y</math>, so this means showing that <math>x\in U</math> such that <math>x\in X</math> or <math>x\in Y</math>. Since <math>X\subseteq U</math>, we have that <math>x\in U</math> and by definition of <math>x</math> we know that <math>x\in X</math>. Thus the “or” statement is true and hence <math>x\in X\cup Y</math>. This shows that <math>X\subseteq X\cup Y</math>.

File:Union.png

2015-06-29T20:04:12Z

Scott Roby1:

Unions

2015-06-29T20:00:36Z

Scott Roby1: Created page with "== Definition == Let <math>X</math> and <math>Y</math> be subsets of some universal set <math>U</math>. The '''''union of <math>X</math> and <math>Y</math>''''', written <math..."

== Definition ==
Let <math>X</math> and <math>Y</math> be subsets of some universal set <math>U</math>. The '''''union of <math>X</math> and <math>Y</math>''''', written <math>X\cup Y</math>, is the set of all <math>x</math> in <math>U</math> which are in at least one of the sets <math>X</math> or <math>Y</math>.<br />
Symbolically, <math>X\cup Y=\lbrace x\in U : x\in X \text{ or } x\in Y\rbrace</math>.

== Examples ==
=== Example 1 ===
Determine the union of the sets <math>X=\lbrace 1,2,5,8 \rbrace</math> and <math>Y=\lbrace 1,3,6,9 \rbrace</math>.

'''Solution''' By definition, we wish to find the set of all elements which are in at least one of the two sets. Thus, we will collect all unique elements into a set as long as they appear once or more between the sets. Our solution is <math>X\cup Y=\lbrace 1,2,3,5,6,8,9 \rbrace</math>.

=== Example 2 ===
Prove that for any sets <math>X</math> and <math>Y</math>, <math>X\subseteq X\cup Y</math>.

'''Proof''' Let <math>x\in X</math>. We wish to show that <math>x\in X\cup Y</math>, so this means showing that <math>x\in U</math> such that <math>x\in X</math> or <math>x\in Y</math>. Since <math>X\subseteq U</math>, we have that <math>x\in U</math> and by definition of <math>x</math> we know that <math>x\in X</math>. Thus the “or” statement is true and hence <math>x\in X\cup Y</math>. This shows that <math>X\subseteq X\cup Y</math>.

Math 144

2015-06-29T19:56:31Z

Scott Roby1: /* Pages In Progress */

== General Resources ==

books, blurbs etc

== Pages In Progress ==

[[Truth Tables]]

[[Subsets]]

[[Unions]]

== Pages For Review ==

== Exams ==

== Files ==
any uploaded file relating to set theory goes here

Subsets

2015-06-29T19:44:08Z

Scott Roby1:

== 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> .

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>.

== Writing Proofs ==

'''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:

'''Proof''' Let <math>x\in X</math>. ''(logical argument)'', thus <math>x\in Y</math>. This shows that <math>X\subseteq Y</math>.

=== Remark ===

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>.

The following is a write-up of the solution of Example 1 as a formal proof:

'''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>.

File:Subset.png

2015-06-29T19:39:27Z

Scott Roby1:

Subsets

2015-06-29T19:36:59Z

Scott Roby1: /* Writing Proofs */

== 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> .

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>.

== Writing Proofs ==

'''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:

'''Proof''' Let <math>x\in X</math>. ''(logical argument)'', thus <math>x\in Y</math>. This shows that <math>X\subseteq Y</math>.

=== Remark ===

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>.

The following is a write-up of the solution of Example 1 as a formal proof:

'''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>.

Subsets

2015-06-29T19:32:16Z

Scott Roby1:

== 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> .

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>.

== Writing Proofs ==

'''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>.

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>.

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>.

Subsets

2015-06-28T03:01:22Z

Scott Roby1: /* Definition */

Subsets

2015-06-28T02:27:02Z

Scott Roby1:

Subsets

2015-06-28T02:26:19Z

Scott Roby1:

== 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> .

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>.

Subsets

2015-06-28T02:25:31Z

Scott Roby1: 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..."

== 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>.

Math 144

2015-06-28T02:18:18Z

Scott Roby1: /* Pages In Progress */

== General Resources ==

books, blurbs etc

== Pages In Progress ==

[[Truth Tables]]

[[Subsets]]

== Pages For Review ==

== Exams ==

== Files ==
any uploaded file relating to set theory goes here

Math 144

2015-06-28T02:17:15Z

Scott Roby1: /* Pages In Progress */

== General Resources ==

books, blurbs etc

== Pages In Progress ==

[[Truth Tables]]
[[Subsets]]

== Pages For Review ==

== Exams ==

== Files ==
any uploaded file relating to set theory goes here