User:IssaRice/Stringing together relations and binary operations - Revision history

IssaRice: IssaRice moved page User:IssaRice/Stringing together relations to User:IssaRice/Stringing together relations and binary operations

2019-08-04T03:59:56Z

IssaRice moved page User:IssaRice/Stringing together relations to User:IssaRice/Stringing together relations and binary operations

← Older revision	Revision as of 03:59, 4 August 2019
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IssaRice at 04:24, 3 August 2019

2019-08-03T04:24:55Z

← Older revision		Revision as of 04:24, 3 August 2019
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	<math>x \in A \times B</math> means <math>x \in (A\times B)</math>, not <math>\{T\}\times B</math> or <math>\{F\}\times B</math>. The interpretation "<math>x \in A</math> and <math>A \times B</math>" does not even parse since <math>A \times B</math> is not a statement.		<math>x \in A \times B</math> means <math>x \in (A\times B)</math>, not <math>\{T\}\times B</math> or <math>\{F\}\times B</math>. The interpretation "<math>x \in A</math> and <math>A \times B</math>" does not even parse since <math>A \times B</math> is not a statement.


			<math>n\cdot m \mid k</math>

IssaRice at 03:45, 3 August 2019

2019-08-03T03:45:57Z

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	On the other hand, if <math></math> is some binary operation on a set <math>S</math>, and <math>a,b,c \in S</math>, then <math>abc</math> means <math>(ab)c</math> if <math></math> associates to the left and means <math>a(bc)</math> if <math></math> associates to the right. If <math></math> is associative, then these two are the same, so <math>abc</math> means either/both.		On the other hand, if <math></math> is some binary operation on a set <math>S</math>, and <math>a,b,c \in S</math>, then <math>abc</math> means <math>(ab)c</math> if <math></math> associates to the left and means <math>a(bc)</math> if <math></math> associates to the right. If <math></math> is associative, then these two are the same, so <math>abc</math> means either/both.

			A relation has type <math>S \times S \to \{T,F\}</math>, and a binary operation has type <math>S\times S \to S</math>. Things can get confusing when we take <math>S = \{T,F\}</math>, because now there are two possible interpretations (depending on whether we take the relation one or the binary operation one).

	One interesting exception to this is when we use <math>\implies</math> between propositions. Let <math>p,q,r</math> be three propositions. What does <math>p \implies q \implies r</math> mean? Some possibilities are:		One interesting exception to this is when we use <math>\implies</math> between propositions. Let <math>p,q,r</math> be three propositions. What does <math>p \implies q \implies r</math> mean? Some possibilities are:
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	* <math>p \implies (q \implies r)</math>		* <math>p \implies (q \implies r)</math>
	* <math>(p \implies q) \implies r</math>		* <math>(p \implies q) \implies r</math>

			other examples to look at:

			<math>x \in A \times B</math> means <math>x \in (A\times B)</math>, not <math>\{T\}\times B</math> or <math>\{F\}\times B</math>. The interpretation "<math>x \in A</math> and <math>A \times B</math>" does not even parse since <math>A \times B</math> is not a statement.

IssaRice at 03:35, 3 August 2019

2019-08-03T03:35:11Z

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	In fact, the relation that is used does not have to be the same in both places. We might write <math>p \in B \subseteq U</math> to mean "<math>p \in B</math> and <math>B \subseteq U</math>".		In fact, the relation that is used does not have to be the same in both places. We might write <math>p \in B \subseteq U</math> to mean "<math>p \in B</math> and <math>B \subseteq U</math>".

	On the other hand, if <math></math> is some binary operation on a set <math>S</math>, and <math>a,b,c \in S</math>, then <math>abc</math> means <math>(ab)c</math> if <math></math> associates to the left and means <math>a(bc)</math> if <math>*</math> associates to the right.		On the other hand, if <math></math> is some binary operation on a set <math>S</math>, and <math>a,b,c \in S</math>, then <math>abc</math> means <math>(ab)c</math> if <math></math> associates to the left and means <math>a(bc)</math> if <math></math> associates to the right. If <math></math> is associative, then these two are the same, so <math>abc</math> means either/both.

	One interesting exception to this is when we use <math>\implies</math> between propositions. Let <math>p,q,r</math> be three propositions. What does <math>p \implies q \implies r</math> mean? Some possibilities are:		One interesting exception to this is when we use <math>\implies</math> between propositions. Let <math>p,q,r</math> be three propositions. What does <math>p \implies q \implies r</math> mean? Some possibilities are:

IssaRice at 03:34, 3 August 2019

2019-08-03T03:34:32Z

← Older revision		Revision as of 03:34, 3 August 2019
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	In fact, the relation that is used does not have to be the same in both places. We might write <math>p \in B \subseteq U</math> to mean "<math>p \in B</math> and <math>B \subseteq U</math>".		In fact, the relation that is used does not have to be the same in both places. We might write <math>p \in B \subseteq U</math> to mean "<math>p \in B</math> and <math>B \subseteq U</math>".

			On the other hand, if <math></math> is some binary operation on a set <math>S</math>, and <math>a,b,c \in S</math>, then <math>abc</math> means <math>(ab)c</math> if <math></math> associates to the left and means <math>a(bc)</math> if <math>*</math> associates to the right.

	One interesting exception to this is when we use <math>\implies</math> between propositions. Let <math>p,q,r</math> be three propositions. What does <math>p \implies q \implies r</math> mean? Some possibilities are:		One interesting exception to this is when we use <math>\implies</math> between propositions. Let <math>p,q,r</math> be three propositions. What does <math>p \implies q \implies r</math> mean? Some possibilities are:

IssaRice at 03:32, 3 August 2019

2019-08-03T03:32:59Z

← Older revision		Revision as of 03:32, 3 August 2019
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	In fact, the relation that is used does not have to be the same in both places. We might write <math>p \in B \subseteq U</math> to mean "<math>p \in B</math> and <math>B \subseteq U</math>".		In fact, the relation that is used does not have to be the same in both places. We might write <math>p \in B \subseteq U</math> to mean "<math>p \in B</math> and <math>B \subseteq U</math>".

			One interesting exception to this is when we use <math>\implies</math> between propositions. Let <math>p,q,r</math> be three propositions. What does <math>p \implies q \implies r</math> mean? Some possibilities are:

			* <math>p \implies q</math> and <math>q \implies r</math>
			* <math>p \implies (q \implies r)</math>
			* <math>(p \implies q) \implies r</math>

IssaRice at 03:30, 3 August 2019

2019-08-03T03:30:48Z

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	For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.		For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.

			Another example is given sets <math>A,B,C</math> we can write <math>A \subseteq B \subseteq C</math> or <math>A \supseteq B \supseteq C</math>.

			In fact, the relation that is used does not have to be the same in both places. We might write <math>p \in B \subseteq U</math> to mean "<math>p \in B</math> and <math>B \subseteq U</math>".

IssaRice: Undo revision 2331 by IssaRice (talk)

2019-08-03T03:29:06Z

Undo revision 2331 by IssaRice (talk)

← Older revision		Revision as of 03:29, 3 August 2019
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	If <math>R</math> is a relation on a set <math>X</math>, and <math>x,y,z</math> are elements of <math>X</math>, we sometimes write <math>x \~~mathbin~~{R} y \~~mathbin~~{R} z</math> as an abbreviation of "<math>x \~~mathbin~~{R} y</math> and <math>y \~~mathbin~~{R} z</math>. This makes sense especially when <math>R</math> is a transitive relation, because in that case we also have <math>x \~~mathbin~~{R} z</math>, which is suggested by the notation "<math>x \~~mathbin~~{R} y \~~mathbin~~{R} z</math>".		If <math>R</math> is a relation on a set <math>X</math>, and <math>x,y,z</math> are elements of <math>X</math>, we sometimes write <math>x \mathrel{R} y \mathrel{R} z</math> as an abbreviation of "<math>x \mathrel{R} y</math> and <math>y \mathrel{R} z</math>. This makes sense especially when <math>R</math> is a transitive relation, because in that case we also have <math>x \mathrel{R} z</math>, which is suggested by the notation "<math>x \mathrel{R} y \mathrel{R} z</math>".

	For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.		For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.

IssaRice at 03:28, 3 August 2019

2019-08-03T03:28:48Z

← Older revision		Revision as of 03:28, 3 August 2019
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	If <math>R</math> is a relation on a set <math>X</math>, and <math>x,y,z</math> are elements of <math>X</math>, we sometimes write <math>x \~~mathrel~~{R} y \~~mathrel~~{R} z</math> as an abbreviation of "<math>x \~~mathrel~~{R} y</math> and <math>y \~~mathrel~~{R} z</math>. This makes sense especially when <math>R</math> is a transitive relation, because in that case we also have <math>x \~~mathrel~~{R} z</math>, which is suggested by the notation "<math>x \~~mathrel~~{R} y \~~mathrel~~{R} z</math>".		If <math>R</math> is a relation on a set <math>X</math>, and <math>x,y,z</math> are elements of <math>X</math>, we sometimes write <math>x \mathbin{R} y \mathbin{R} z</math> as an abbreviation of "<math>x \mathbin{R} y</math> and <math>y \mathbin{R} z</math>. This makes sense especially when <math>R</math> is a transitive relation, because in that case we also have <math>x \mathbin{R} z</math>, which is suggested by the notation "<math>x \mathbin{R} y \mathbin{R} z</math>".

	For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.		For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.

IssaRice: Created page with "If $R$ is a relation on a set $X$ , and $x,y,z$ are elements of $X$ , we sometimes write $x \mathrel{R} y \mathrel{R} z$ a..."

2019-08-03T03:28:10Z

Created page with "If <math>R</math> is a relation on a set <math>X</math>, and <math>x,y,z</math> are elements of <math>X</math>, we sometimes write <math>x \mathrel{R} y \mathrel{R} z</math> a..."

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If <math>R</math> is a relation on a set <math>X</math>, and <math>x,y,z</math> are elements of <math>X</math>, we sometimes write <math>x \mathrel{R} y \mathrel{R} z</math> as an abbreviation of "<math>x \mathrel{R} y</math> and <math>y \mathrel{R} z</math>. This makes sense especially when <math>R</math> is a transitive relation, because in that case we also have <math>x \mathrel{R} z</math>, which is suggested by the notation "<math>x \mathrel{R} y \mathrel{R} z</math>".

For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.