Tagged Questions

For reflexive, symmetric and transitive relations. Use it with the tag (relations).

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Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$

Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$. Well, I've proved really easily that it is reflexsive, symmetrical and transitive. But I'm ...
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Give a complete set of equivalence class representatives for an equivalence relation on the natural numbers (including zero)

The full question: Having an equivalence relation $\sim$ on $\Bbb N$ defined by: $a \sim b$ meaning $a,b\in\Bbb N$ such that $a=b*10^k,$ for some $k\in\Bbb Z$, give a complete set of equivalence ...
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Relation $(x,y) \in \rho \Leftrightarrow (\exists k \in \mathbb{Z})\mid x- y=3k$

I know that there is a similar question here, but it's about classes of equivalence of this relation. I would like to know how to prove that this is an equivalence relation. It seems simple, but the ...
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congruent modulo S

This is most likely a silly English question, but in Roman's "Advanced Linear Algebra," on page 21, he writes that: Let $S$ be a subset of a commutative ring $R$ with identity. Let $\equiv$ be the ...
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relations and equivalence classes

$S$ is the set of all equivalence relations on set $A = \{1,2,3,4,5,6,7,8,9\}$ in which one of the equivalence classes is $\{1,3,5,7,9\}$. What is the size of $S$? I don't even know how to start... ...
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Propositional Equivalence

Are the following two propositions equivalent? p IMPLIES (q IMPLIES r) p IMPLIES (q AND r) From what I can tell, using the logical equivalences, this should be false, correct? p ...
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What is the meaning of cyclically equivalence classes of multiple indices?

Can you please give me an example for the cyclic equivalence classes of multiple indices on the following set? $$I(w,d)=\{\ (k_1,...k_d)\mid k_1+\ldots+k_d=w, k_1,...k_d \ge 1\,\}$$ where $w$: ...
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For each $x \in X$, $x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$? Shouldn't it be for each $y \in X$?

"Definition 6. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. For each $x \in X$, we define $x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$ which is called the equivalence ...
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Where in (a) and (b) above shows that $x/\mathscr E = y/\mathscr E \Rightarrow x \mathscr E y$?

"Definition 6. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. For each $x \in X$, we define $x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$ which is called the equivalence class ...
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Equivalence relation on $S^2$ and unit square make same space

I think I understand a bit of this task, but I hope someone can look critical to my answers: Let $X$ be the space obtained from the sphere $S^2$ by gluing the north and the south pole (with the ...
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Canonical injection is compatible with induced equivalence relation?

Let $R$ be an equivalence relation on a set $E$, let $A$ be a subset of $E$, let $j : A \to E$ be the canonical injection, and let $R_A$ be the equivalence relation induced by $R$ on $A$ (that is, the ...
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Relations consisted of triples

A relation $m$ is defined on the set of nonnegative real triples as follows: $(a_1,a_2,a_3)\,m\, (b_1,b_2,b_3)$ if two of the inequalities $a_1>b_1,a_2>b_2, a_3>b_3$ are satisfied. ...
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Does symmetric property of Equivalence hold when the left hand side of the equation is in indeterminate form?

The Symmetric Property of Equivalence is a=b implies b=a. This property does not have any conditions on a or b. But what if through manipulation I get $a=\frac00$ where $a$ is a real number does this ...
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Find the equivalence classes for $a T b \iff \frac a b \in \Bbb Q$

Given the set $S = \{ x − \sqrt 5 y : x,y \in \Bbb Q, \ x − \sqrt 5 y \ne 0 \}$, assume the relation $T$ is defined on $S$ by $a T b \iff \frac a b \in \Bbb Q$. How can I find the distinct ...
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$x\sim y$ if $|x-y|\le 3$, then is $\sim$ or R an equivalence relation?

Let R or $\sim$ be the relation defined on Z by $$x\sim y\text{ if } |x-y| \le 3$$ Is $\sim$ an equivalence relation? It is reflexive and symmetric if I did it correctly. However, I am having doubt ...
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Equivalence relation and Distinct equivalence classes

Given the set $S = \{x-y \sqrt5: x, y$ are rational numbers and $x-y \sqrt5 \neq 0\}$. Assume the relation $T$ is de fined on the set $S$ by $a T b$ if $a/b$ is a rational number. Question has two ...
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Finding distinct equivalence classes

Q: Given the set $S = \{x - y\sqrt 5 : \text{x, y are rational numbers and }x - y \sqrt5 \neq 0 \}$. Assume the relation defined on the set $S$ by $a\ T\ b$ if $a/b$ is a rational number. Find the ...
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Equivalence relation: $aRb$ iff $2a+3b$ is divisible by $5$

Prove that $R$ is an equivalence relation: $aRb$ iff $2a+3b$ is divisible by $5$. Here $a,b\in \mathbb{Z}$ (set of integers). I can prove that $R$ is reflexive and transitive. How to prove it's ...
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E1 and E2 are equivalent then they are “almost equivalent”

Given : 2 statements E1, E2 in relational algebra are "almost equivalent" if every phase in the database D ,except finite number of D's E1(D)=E2(D). E(D) means the result of activating the statement E ...
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What is the symbol for “coincident” in geometry?

I am looking for a symbol to say that one geometrical figure coincides with another without writing the phrase "is coincident with." For example, the altitude $a$ of an equilateral triangle coincides ...
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Finding distinct equivalence classes.

I am going through some practice questions and am having trouble to finding distinct equivalence classes and this is my understanding so far. Let S be a nonempty subset of Z, and let R be a relation ...
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Can someone help me understand as to why the following are equivalent when x is a bound variable that does not occur free in A? $\forall x (A \lor B) \iff A \lor \forall x B$ $\exists x (A \... 1answer 20 views On an exercise that asks for a homeomorphism between a quotient space and a metrizable space. I have the solution to the exercise but have a doubt on one thing, I state the exercise: Given $$X = \{ (x,y) \in R^2 | x = \frac{1}{n}, n \in N \}$$ and$Y = X/_{\sim}$where the equivalence ... 2answers 40 views Partition into “fibers”$f^{-1}(y) \in Y$Consider any surjective map f from a set X onto another set Y. We can define an equivelance relation on X by$x_1Rx_2$if$ f(x_1)=f(x_2)$. Check that this is an equivelance relation. Show that the ... 0answers 68 views Given$S=\{0,1,2,3,4,5\}$, ﬁnd the partition induced by the equivalence relation$R$I am currently taking discrete math and have been given the following question to answer. Given$S=\{0,1,2,3,4,5\}$, ﬁnd the partition induced by the equivalence relation$R$where$R=\{(0,0),(0,4),(...
I'm looking at the following equivalence relation on $\mathbb{Z}$: $a \sim b$ if and only if there exist $n,m \in \mathbb{N}_{>0}$ so that $a^n = b^m$ I'm trying to determine what the cardinality ...
can anyone help me on this problem? I have the set $\{0,1,3,8,9\}$ and I want to define an example of an equivalence relation. I know that to be an equivalence relation it needs to be reflexive, ...