Explanation of Relations, Ordered Pairs, and Sets | Mathematics | Page 1

tribal bison Sep 11, 2022, 1:58 AM

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Recently started learning about these concepts and am very lost.

spring canopy Sep 11, 2022, 2:11 AM

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I'd like to buy a vowel

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Do you have a specific question or...?

tribal bison Sep 11, 2022, 2:12 AM

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More like clarification lol

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So from my understanding:

Ordered pairs are pairs of elements that must follow a certain order; i.e (a, b) != (b, a)

Sets are collections of Ordered Pairs

Relations are collections of Sets?

spring canopy Sep 11, 2022, 2:14 AM

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sets are things that has things in them

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like {cat, 17, pi, [1,1,1]} is a set

tribal bison Sep 11, 2022, 2:14 AM

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ohhh

spring canopy Sep 11, 2022, 2:14 AM

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ordered pairs are just.. pairs of things, with fixed order

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namely given two sets $A$ and $B$, I define the following set $A\times B:={(a,b)|a\in A\text{and}b\in B}$

bitter bloomBOT Sep 11, 2022, 2:15 AM

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Omegabet_

spring canopy Sep 11, 2022, 2:16 AM

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aka the set of all ordered pairs where the 1st thing is from A and the 2nd thing is from B

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common example you're use to, the xy plane is $\mathbb{R}\times\mathbb{R}=:\mathbb{R}^2$

regal oak Sep 11, 2022, 2:16 AM

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And sets are just lists of things

bitter bloomBOT Sep 11, 2022, 2:16 AM

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Omegabet_

tribal bison Sep 11, 2022, 2:16 AM

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hold on man im not too caught up with this math syntax lol

spring canopy Sep 11, 2022, 2:16 AM

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spring canopy sets are things that has things in them

~~you really need to read what I say lol~~

tribal bison Sep 11, 2022, 2:17 AM

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could you put it into layman's terms please

spring canopy Sep 11, 2022, 2:17 AM

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you'd need to explain what you're confused on

tribal bison Sep 11, 2022, 2:17 AM

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basically every symbol you included in your explanation that isn't part of the alphabet lol

spring canopy Sep 11, 2022, 2:17 AM

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{} means set, $\in$ means "in" $|$ means "such that"

bitter bloomBOT Sep 11, 2022, 2:17 AM

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Omegabet_

spring canopy Sep 11, 2022, 2:18 AM

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$\mathbb{R}$ is the real numbers

bitter bloomBOT Sep 11, 2022, 2:18 AM

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Omegabet_

tribal bison Sep 11, 2022, 2:19 AM

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hmm okay

spring canopy Sep 11, 2022, 2:20 AM

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but anyway, a relation is then just a subset of this AxB set

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namely all pairs of elements that satisfy the relation

tribal bison Sep 11, 2022, 2:23 AM

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sorry im having a really hard time understanding your explanation of sets. i understand that they are just structures to hold elements within, but could you give me another example that might be easier to understand?

spring canopy Sep 11, 2022, 2:25 AM

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Not really tbh

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Set is just a list of stuff

tribal bison Sep 11, 2022, 2:26 AM

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well what would you consider the most common usage of a set

spring canopy Sep 11, 2022, 2:26 AM

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Probability tbh

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But honestly everything in math uses sets

tribal bison Sep 11, 2022, 2:28 AM

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hmm alright

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could you elaborate on what you meant by relations being a "subset" of sets? the resource im using right now contains the following example as the relation T:
T = {(3, 0), (7, -5),(-5, 5), (3, 6)}

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to me, this looks alot like your explanation of what a set is

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i'm not sure if this needs clarification but to my observation, i believe the elements of T are ordered pairs of plotted points.

elder basin Sep 11, 2022, 5:40 AM

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tribal bison could you elaborate on what you meant by relations being a "subset" of sets? the...

If we're talking set theory, we need to talk about the axioms of set theory. This in particular refers to the axiom of subsets; given any set A, it is possible to construct the set B containing some, but not necessarily all, of the elements of A.

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More precisely, the axiom schema of subsets states that for any set A, it is possible to construct the set B containing all elements x such that x is in B if and only if x is in A and some predicate p holds for x.

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It's an axiom schema because there is a specific form of the axiom for every predicate p.

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We call this the axiom of subsets because B is a subset of A; that is, all of the elements of B are elements of A.

#Explanation of Relations, Ordered Pairs, and Sets