Mathematics/Discrete Math/Relations and Functions - Revision history

Brodriguez: Brodriguez moved page Discrete Math/Relations and Functions to Mathematics/Discrete Math/Relations and Functions

2020-10-25T17:23:37Z

Brodriguez moved page Discrete Math/Relations and Functions to Mathematics/Discrete Math/Relations and Functions

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Revision as of 17:23, 25 October 2020

Brodriguez: Brodriguez moved page Discrete Math - Relations and Functions to Discrete Math/Relations and Functions: Clean url with subpages

2020-05-15T08:55:17Z

Brodriguez moved page Discrete Math - Relations and Functions to Discrete Math/Relations and Functions: Clean url with subpages

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Revision as of 08:55, 15 May 2020

Brodriguez: Create equivalence relations section

2019-09-10T17:30:23Z

Create equivalence relations section

← Older revision		Revision as of 17:30, 10 September 2019
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	A function has an '''inverse'' iff it's bijective. We can then define the inverse as <code>f^-1(b)=a iff f(a)=b</code>		A function has an '''inverse'' iff it's bijective. We can then define the inverse as <code>f^-1(b)=a iff f(a)=b</code>

			== Equivalence Relations ==
			The following definitions refer to some <code>a, b, c ∈ S</code> for some arbitrary set S.

			Also recall that, fore each paring (a, b), we can denote that they're related with <code>a R b</code>.

			* '''Reflexive Relation''' - Defined as <code>a = a</code>.
			** Effectively, what this means is that every element is has a relation/paring to itself.
			** Ex: For <code>a, b, c ∈ S</code>, we must have some <code>(a, a), (b, b), and (c, c)</code> to be reflexive.
			** Ex: The statements "is married to" or "is a coworker of".
			* '''Symmetric Relation''' - If <code>a = b</code>, then <code>b = a</code>.
			** Ex: For <code>a, b, c ∈ S</code>, if we have have some <code>(a, b)</code>, then we must also have <code>(b, a)</code> to be symmetric.
			** Ex: The numbers 3 and 8 when applied to a modulus of 5.
			* '''Transitive Relation''' - If <code>a = b</code> and <code>b = c</code>, then <code>a = c</code>.
			** Ex: For <code>a, b, c ∈ S</code>, if we have some <code>(a, b) and (b, c)</code>, then we must also have <code>(a, c)</code> to be transitive.
			* '''Equivalence Relation''' - A binary relation that is reflexive, symmetric, and transitive all at once.
			* '''Irreflexive'''/'''Anti-Reflexive''' - A binary relation in which no elements relate to themselves.

Brodriguez: Add function properties

2019-09-10T16:30:44Z

Add function properties

← Older revision		Revision as of 16:30, 10 September 2019
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	Recall from [[Discrete Math - Sets]] (basic set operations) that a '''Cartesian Product''' is a mapping of ordered pairs from sets A={a1, a2, ...} and B={b1, b2, ...} such that		Recall from [[Discrete Math - Sets]] (basic set operations) that a '''Cartesian Product''' is a mapping of all possible ordered pairs from sets A={a1, a2, ...} and B={b1, b2, ...} such that
	<code>A x B={(a1, b1), (a1, b2), ..., (a2, b1), (a2, b2), ...}</code>.		<code>A x B={(a1, b1), (a1, b2), ..., (a2, b1), (a2, b2), ...}</code>.

	Using this fact, we can define some terms:		Using this fact, we can define some terms:

			== Function Definitions ==
	* '''Binary Relation''' - A subset of the cartesian product <code>A x B</code> for some arbitrary, non-zero sets ''A'' and ''B''.		* '''Binary Relation''' - A subset of the cartesian product <code>A x B</code> for some arbitrary, non-zero sets ''A'' and ''B''.
	** When an element ''a ∈ A'' is paired with an element ''b ∈ B'' through a binary relation, we can denote that they're related with <code>a R b</code>.		** When an element ''a ∈ A'' is paired with an element ''b ∈ B'' through a binary relation, we can denote that they're related with <code>a R b</code>.
	* '''Function''' - A binary relation such that every element in ''A'' corresponds to, at most, one element in ''B''.		* '''Function''' - A binary relation such that every element in ''A'' corresponds to, at most, one element in ''B''.
	* '''Domain''' - Given all <code>{(a, b) ∈ f \| a ∈ A, b ∈ B}</code>, the "domain" is the full set ''A''.		* '''Domain''' - Given all possible <code>{(a, b) ∈ f \| a ∈ A, b ∈ B}</code>, the "domain" is the full set ''A''.
	* '''Codomain''' - Given all <code>{(a, b) ∈ f \| a ∈ A, b ∈ B}</code>, the "codomain" is the full set ''B''.		* '''Codomain''' - Given all possible <code>{(a, b) ∈ f \| a ∈ A, b ∈ B}</code>, the "codomain" is the full set ''B''.
	* '''Argument''' - Given all <code>{(a, b) ∈ f \| a ∈ A, b ∈ B}</code>, each ''a'' can be considered an "argument".		* '''Argument''' - Given all possible <code>{(a, b) ∈ f \| a ∈ A, b ∈ B}</code>, each ''a'' can be considered an "argument".
	* '''Value''' - Given all <code>{(a, b) ∈ f \| a ∈ A, b ∈ B}</code>, each ''b'' can be considered a "value".		* '''Value''' - Given all possible <code>{(a, b) ∈ f \| a ∈ A, b ∈ B}</code>, each ''b'' can be considered a "value".
	* '''Finite Sequence''' - A function that has at most ''n'' domain → codomain pairings.		* '''Finite Sequence''' - A function that has at most ''n'' domain → codomain pairings.
	** In simpler terms, a function that has a limited/set number of a → b mappings.		** In simpler terms, a function that has a limited/set number of a → b mappings.
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	Note that two functions are considered equivalent if they have the same domain and codomain.		Note that two functions are considered equivalent if they have the same domain and codomain.

			== Function Properties ==
			* '''Image''' - The set of all output values a function may have.
			* '''Preimage'''/'''Inverse Image''' - The set of all elements in a domain that map to the codomain.
			* '''Injection''' - "One-to-one". Aka, a function where each element in the codomain is mapped to by, at most, one element of the domain.
			** To put it another way, each input provides exactly one output.
			* '''Surjection''' - "Onto". Aka, a function where each element of the codomain is mapped to by, at least, one element of the domain.
			** To put it another way, each possible output corresponds to at least one input.
			* '''Bijection''' - "One-to-one Correspondence". Aka, a function that is both Injective and Surjective at the same time.
			** Aka, each input relates to exactly one output, all possible outputs correspond to exactly one input each.

			A function has an '''inverse'' iff it's bijective. We can then define the inverse as <code>f^-1(b)=a iff f(a)=b</code>

Brodriguez: Create page

2019-09-10T08:24:30Z

Create page

New page

Recall from [[Discrete Math - Sets]] (basic set operations) that a '''Cartesian Product''' is a mapping of ordered pairs from sets A={a1, a2, ...} and B={b1, b2, ...} such that
<code>A x B={(a1, b1), (a1, b2), ..., (a2, b1), (a2, b2), ...}</code>.

Using this fact, we can define some terms:
* '''Binary Relation''' - A subset of the cartesian product <code>A x B</code> for some arbitrary, non-zero sets ''A'' and ''B''.
** When an element ''a ∈ A'' is paired with an element ''b ∈ B'' through a binary relation, we can denote that they're related with <code>a R b</code>.
* '''Function''' - A binary relation such that every element in ''A'' corresponds to, at most, one element in ''B''.
* '''Domain''' - Given all <code>{(a, b) ∈ f | a ∈ A, b ∈ B}</code>, the "domain" is the full set ''A''.
* '''Codomain''' - Given all <code>{(a, b) ∈ f | a ∈ A, b ∈ B}</code>, the "codomain" is the full set ''B''.
* '''Argument''' - Given all <code>{(a, b) ∈ f | a ∈ A, b ∈ B}</code>, each ''a'' can be considered an "argument".
* '''Value''' - Given all <code>{(a, b) ∈ f | a ∈ A, b ∈ B}</code>, each ''b'' can be considered a "value".
* '''Finite Sequence''' - A function that has at most ''n'' domain → codomain pairings.
** In simpler terms, a function that has a limited/set number of a → b mappings.
* '''Infinite Sequence''' - A function that has a non-terminating/infinitely extending number of domain → codomain pairings.

Note that two functions are considered equivalent if they have the same domain and codomain.