#reflexive symmetric transitive relations

[edusquaremaths]

Q10 Exercise1.1 I Class 12 Maths NCERT Chapter 1 Relations and Functions | NCERT solutions

NCERT Class 12Chapter: Relations and FunctionsExercise 1.1Question 10: Give an example of a relation. Which is(i) Symmetric but neither reflexive nor transitive.(ii) Transitive but neither reflexive nor symmetric.(iii) Reflexive and symmetric but not transitive.(iv) Reflexive and transitive but not symmetric.(v) Symmetric and transitive but not reflexive.

[maybeasunflower]

When I see "transitive", I can't help think of equivalence relations (a maths thing), which I know are defined by three properties: they are transitive (A~B and B~C => A~C), reflexive (A~B => B~A), and the other one (A~A is true), but I can't remember the name of the other one.

Then I look up equivalence relations and find the "other one" is actually "reflexive", what I think is called "reflexive" is actually called "symmetric", and it's fine.

if shes your girl then why have i slowly been replacing her parts until there’s nothing left of her original body? is she then still your girl?

[bubbloquacious]

A partial equivalence relation is a reflexive, symmetric, and transitive binary relation ≡, which allows you to compare some, but not all pairs of elements x, y of the domain. A total equivalence relation is similar, but we also require that for all x, y we have that x ≡ y or y ≡ x.

[noelements-setempty]

Word rhyming is an equivalence relation

Take the definition that two words rhyme if and only if they end with the same sound.

Reflexive: Every word rhymes with itself.

Well, if two words are the same, all their sounds have to match, including the final one, so this point holds.

Symmetric: If A rhymes with B, B rhymes with A.

This one’s really hard to prove, because it’s so obvious. If A rhymes with B, then the final sounds of A and B are the same. They will still be the same if we swap the words around. Please don’t make me explain it more, I’ll cry.

Transitive: If A rhymes with B and B rhymes with C, then A rhymes with C.

Call the sound at the end of word A ‘&’. If A rhymes with B, then B also has to end with ‘&’. If B rhymes with C, and B ends with ‘&’, then C also has to end with ‘&’. This means that both A and C end with ‘&’, and so A rhymes with C.

There we go. The argument no one cares about but me has been made. Rhyme is an equivalence relation. You can all go home.

[lipshits-continuous]

If your relation(ship) is

Reflexive

Symmetric

Transitive

It's not romantic, it's an equivalence

[codeshive]

Homework 3 CSCI 301 solved

Use the method of proof by contradiction. 1. [4 points] Prove that √6 is irrational. 2. [4 points] If �, � �ℤ , then �) − 4� − 2 ≠ 0. 3. [4 points] Suppose Suppose A≠ Ø. Since Ø ⊆A×A, the set R= Ø is a relation on A. Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. 4. [4 points] Define a relation R on Z as xRy if and only if 4 | (x + 3y) Prove R is an equivalence…

[codingprolab]

Homework 3 CSCI 301

Use the method of proof by contradiction. 1. [4 points] Prove that √6 is irrational. 2. [4 points] If �, � �ℤ , then �) − 4� − 2 ≠ 0. 3. [4 points] Suppose Suppose A≠ Ø. Since Ø ⊆A×A, the set R= Ø is a relation on A. Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. 4. [4 points] Define a relation R on Z as xRy if and only if 4 | (x + 3y) Prove R is an equivalence…

[class12maths]

Relations and Functions: Class 12 Maths Video Tutorials — Mathyug

Are you a Class 12 student gearing up for your Maths exams? Do the concepts of relations and functions seem daunting? Fret not! Ashish Sir from MathYug has designed an exceptional self-study course that covers everything you need to master these topics. This comprehensive course includes video tutorials, detailed notes, and assignments that ensure thorough understanding and practice.

What You’ll Learn

Ashish Sir’s course on MathYug is meticulously structured to provide a deep understanding of relations and functions, a crucial component of Class 12 Mathematics. The course is divided into two main parts:

1. Relations

In this segment, you will explore various types of relations. Here’s a brief overview of what you’ll learn:

Empty Relations: Understanding the concept of relations where no element of a set is related to any element of another set.

Universal Relations: Learning about relations where every element of a set is related to every element of another set.

Trivial Relations: Delving into the simplest form of relations that only relate to the identical pairs.

Reflexive Relations: Understanding relations where every element is related to itself.

Symmetric Relations: Exploring relations where if one element is related to another, the second element is also related to the first.

Transitive Relations: Learning about relations where if one element is related to a second, and the second to a third, then the first is related to the third.

Equivalence Relations: Discovering relations that are reflexive, symmetric, and transitive.

Equivalence Classes: Understanding the partition of a set into disjoint subsets where each subset is an equivalence class.

To ensure comprehensive understanding, the course provides practice questions from diverse sources such as NCERT Textbook exercises, NCERT Examples, Board’s Question Bank, RD Sharma, and NCERT Exemplar. You can download the PDF of assignments within the course for additional practice.

2. Functions

The second part of the course focuses on functions, including:

One to One and Onto Functions: Understanding functions where each element of one set is paired with a unique element of another set, and functions that map elements from one set onto every element of another set.

Composite Functions: Learning how to combine two functions to form a new function.

Inverse of a Function: Exploring how to find the function that reverses the effect of the original function.

Similar to the relations segment, this part also includes practice questions from various reputable sources. The assignments are designed to reinforce your understanding and are available for download within the course.

Sample Videos

To give you a glimpse of the quality of instruction, Ashish Sir has shared sample videos covering some of these essential topics. Watching these will help you understand his teaching methodology and the depth of content provided.

If you’re aiming to excel in your Class 12 Maths exams, Ashish Sir’s course on MathYug is your go-to resource. With detailed video tutorials, comprehensive notes, and extensive assignments, you will gain a solid understanding of relations and functions. Start your journey towards mastering these crucial topics today!

For more information and to access the course, visit MathYug. Happy learning!

[mathyug]

Unlock a deeper understanding of Reflexive, Symmetric, Transitive, and Equivalence Relations, and master Exercise 1.1 in Relations and Functions Class 12 Maths with expert instructor Ashish Sir on MathYug.Explore additional video lectures on our website: https://mathyug.com/class-12-maths

[dorothytheexplorothy]

alternatively always use the same X in the same font, but make sure it's clear from context which one you're referring to.

Definition. two subsets X,X ⊆ X of the natural numbers are called equivalent mod finite, denoted X=*X, if their symmetric difference XΔX is finite, i.e. if there exists a natural number X such that |XΔX| = X.

Proposition. equivalence mod finite is an equivalence relation.

proof. reflexivity holds by ∀X.|XΔX|=0; symmetry by that of the symmetric difference XΔX=XΔX; and transitivity, by the following "triangle inequality" for sets:

∀X,X,X⊆X.∀X,X∊X.(|XΔX|=X∧|XΔX|=X)⇒|XΔXΔX|≤X+X. ▢

Lemma. the finite sets of natural numbers form an equivalence class mod finite.

proof. for any subset X of the naturals, we have XΔX=X, where X is the empty set; therefore, XΔX is finite (i.e. X is equivalent to the empty set) exactly when X is finite. by the properties of equivalence relations any two subsets X,X are equivalent iff there's a third subset equivalent to both, and so when X is finite, X=*X iff X is also finite, since just then they're both equivalent to the empty set. ▢

Proposition. the class of finite sets of natural numbers is the only class with a least element under the usual ordering of sets by inclusion.

proof. the least element in the class of finite sets is the empty set, since it's a subset of every set. for any other equivalence class, assume by way of contradiction there's a least element X. by assumption, X is not equivalent to the empty set, and in particular is infinite. taking X := X\{minX} (the minimum of every set exists since the natural numbers are well-ordered), we have X⊆X and XΔX={minX}, or in other words X<X and X=*X, contradicting the assumption that X was minimal. ▢

Only represent variables by x in various fonts.

[edusquaremaths]

Q9 Exercise1.1 I Class 12 Maths NCERT Chapter 1 Relations and Functions | NCERT solutions

NCERT Class 12Chapter: Relations and FunctionsExercise 1.1Question 9: Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by(i) R = {(a, b) : |a – b| is a multiple of 4}(ii) R = {(a, b) : a = b}is an equivalence relation. Find the set of all elements related to 1 in each case.

[quietimtryingtostudy]

9. Let W denote the set of words in the English dictionary. Define the relation R by R= {(x, y) ∈ W x W such that x and y have at least one letter in common}. Show that this relation R is reflexive and symmetric, but not transitive.

(i vaguely remember my teacher saying that we don't have word problems from this chapter or smtg. but just in case, ya know?)

Took help from one of my friends for this one- but I understood the entire procedure by the end!

(Yeah, I understand. Thanks for sending it though!!)

[bhargaw1]

Equivalence relation

#equivalencerelations Definition 🙂 An equivalence relation on a set A, is a relation “R” in A which is #reflexive, #symmetric, and #transitive.

Definition 🙂 An equivalence relation on a set A, is a relation “R” in A which is reflexive, symmetric, and transitive. Reflexive relation ARB is a relation in set A, then R is called Reflexive relation R is reflexive, i.e., for all a belongs to R (a, a) also belongs to R a=1 & b=a then (1,1) also belongs to R ⭐⭐⭐⭐ Rating: 4 out of 5. Symmetric relation (mirror) Let R be a relation in set…

[mathhombre]

Pat Ballew adds "Equal The mathematical use of equal means that two things are related in a transitive, symmetric, and reflexive way in relation to some specified properties.  The meaning is rooted in the Latin word aeques for level. Chaucer's 1400 Treatise on the Astrolabe uses the term.  Cajori's History of Mathematical Notation gives many early symbols used in the West, both before and after Recorde wrote the now ubiquitous "=" for the symbol in 1557, but even for another century, many mathematicians used no symbol at all, or used a collection of other symbols.In fact, after 1557, Recorde's symbol  did not appear again in print until 1618.  Viete, writing 1n 1571 used the same symbol for difference (the absolute  value of a-b) and this symbol was widely repeated in this use across Europe. Another early symbol for equality was the script conjunction of ae."

When Was The Equals Sign (=) Invented?

The equals sign (=) was introduced by Robert Recorde in 1557 to denote equality between mathematical expressions.

More: https://riverainventions.com/who-invented-mathematical-symbols/

[wuggen]

An innocuous little math thing that I find really neat for some reason: a partial equality relation (symmetric and transitive) almost implies a full equality relation (symmetric, transitive, and reflexive). If a=b, then by symmetry b=a, and by transitivity a=a, and then you've got reflexivity. The only way to have a partial equality relation that is not a full equality relation is if there exists some element that just doesn't appear in the relation at all

[model-theory]

Found out that symmetric, reflexive relations are called Fuzzys. This is an adorable name and very clever since it's just an equivalence relation w/o transitivity so instead of equivalence classes [x] you get vague classes. Arguably ≈ is a fuzzy:

x≈x, x≈y→y≈x, but ≈ isn't transitive