# Tagged Questions

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

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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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### why are equivalence relations called so?

"an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of ...
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### Equivalence relation and partitions [closed]

Define an equivalence relation on the set R that partitions the real line into subsets of length 1.
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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 ...
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### Relation of equivalence with sgn

Test if the relation $$(x, y)ρ(a, b)\Leftarrow\Rightarrow sgn(y+\pi x) = sgn(b + \pi a)$$ is a relation of equivalence on $R^2$ and if so, determine the quotient set and $C_{(2, \pi)}$. Also, ...
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### why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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### Addition in $R_S$ is well defined

Let $R$ be a commutative ring with $1 \neq 0$ and suppose S is a multiplicatively closed subset of $R \backslash {\{\, 0 \,\} }$ containing no zero divisors. We have the relation ∼ defined on $R × S$ ...
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### A relation that is Reflexive & Transitive but neither an equivalence nor partial order relation

Set $A = \{0,7,1\}$ 1. So for a relation that is reflexive and transitive but neither an equivalence relation nor partial order...Can a relation be both partial order and equivalence? Attempt: ...
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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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### Solve the relation with congruence

On $\Bbb Z$ consider the relation $xRy \Leftrightarrow x-y \not\equiv 0 \mod 3$. Prove (with explanation), whether the relation reflexive, symmetric, antisymmetric transitive is and prove if they are ...
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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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### 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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### $x\sim y$ 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 ...
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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 ...
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### Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$\vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m$$ The dimension of ...
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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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### The relation $a_1 \sim a_2 \iff f(a_1 ) = f(a_2 )$ is an equivalence relation

Suppose a function $f : A → B$ is given. Define a relation $\sim$ on $A$ as follows: $a_1 \sim a_2 \iff f(a_1 ) = f(a_2 )$. Prove that $\sim$ is an equivalence relation on $A$. I know that in ...