# Tagged Questions

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

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### Symmetry of a relation

There is a professor in our University who each year posts some homework for his students (1st years at computer studies) and I am trying to solve it for fun. However, now I got stuck on something ...
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### How many distinct equivalence classes does this equivalence on rationals have?

Let $$A = \{ r\in \mathbb Q \mid \exists p\in \mathbb Z,\text{ and q\in \mathbb Z, with p even and q odd, and r = p/q} \}$$ For example, $A$ contains such $2/9, 16/(-34)$, and $4$. $A$ does ...
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### Let $v,w$ be vectors of some vectorial space $V$. If $v=w$, are they said to be equivalent?

Of course two geometrical vectors are called equivalent if they have the same magnitude, direction and orientation. But what about a generic vectorial space? Does the relation $v=w$ keep this name? I ...
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### Show $\phi (a)=\phi (x)$ iff $a^{-1}x \in N$ iff $aN=xN$

disclaimer: This is not a homework question, it's purely a question to reinforce my understanding: Let $\phi :G \rightarrow H$ be a homomorphism of groups with kernel N. $\forall a,x \in G$ show ...
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### Equivalence classes and equivalent relationship

We define a relation S on the set of all integers by: $nSk$ iff $n^2$ $=$ $k^2$ Decide if S is an equivalence relation. If so, what is the equivalence class of $9$? It can be proven that S is an ...
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### How to show if relation on $\mathbb N\times\mathbb N$ defined $(a,b) \sim (c,d)$ by $ad(b+c)=bc(a+d)$ is transitive?

I can show it is reflexive and symmetric but I don't know how to show transitivity using only the properties of natural numbers (no division).
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### if $x\mathcal R y$ defined by $|x|+|y| =|x+y|$. Is it an equivalence relation?

Reflexive and symmetric can be proved as $|x|+|x|=|x+x|$ hence reflexive and $|y|+|x|=|y+x|$ hence symmetric but how transitive?