# Tagged Questions

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### Is an anti-symmetric and asymmetric relation the same? Are irreflexive and anti reflexive the same?

I don't understand the difference between an anti symmetric and asymmetric relation. From my understanding, it is asymmetric if there is not any element where: if (x,y) (y,x). But what if you have ...
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### Is there way to classify the quantifier rank $m$ first order sentence in first order logic

In its simplest situation, for example, if the signature contains only a binary relation $\sigma$, so the signature $\tau = \{ \sigma \}$, what are the inequivalent classes of all first order ...
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### Use of “for all” in definition of reflexive and symmetric relations.

My book says that a relation R on A is reflexive, if $\ (a,a) \in R, \ for\ every \ a \in A$ symmetric, if $\ (a_1,a_2) \in R \implies (a_2,a_1) \in R,\ for\ all\ a_1,a_2 \in A$ Although I ...
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### Generated equivalence relations in logics

Let $L$ be some logic (FO or stronger which is not important for this purpose). Given a $\tau$-structure $A$ and a formula $\varphi(x_1, \dots x_n) \in L[\tau]$ with free variables $x_1, \dots, x_n$. ...
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### Model and countermodel to $\exists x.\forall y. x<y$ (with $<$ an arbitrary relation)

Can someone please help me with this question. I have been struggling with it for ages and can't quite seem to work it out: Let $<$ be a binary relation symbol that we will write infix. Let ...
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### Is my transitivity proof correct for the relation over $\mathbb{Z} \times \mathbb{Z}$ where $(a,b)R(c,d) \iff (a \le c \lor b \le d)$?

I'm having a hard time developing abstract thinking to solve problems regarding a relation's properties. I've spend quite an absurd amount of time on this one, but I think I finally grasped a bit of ...
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### Proving a relation's inverse's properties by knowing the original's.

I'm getting fairly confused with two exercises related to proving a relation's inverse's properties by knowing the original's. I couldn't do either. Any hint is appreciated. If $R$ is a ...
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### How can I further simplify $(a \le b) \lor (b \le a)$ to prove that it is a tautology?

Over $\mathbb{Z}$, $aRb \iff a \le b \lor a = 3b$. Determine if it is total. I think it is: Have arbitrary elements $a,b \in \mathbb{Z}$. We have to prove that $aRb \lor bRa$, which can be ...
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### Proving that $aRb \iff a = b \lor a = b^2$ is antisymmetric.

Over $\mathbb{N}$, $aRb \iff a = b \lor a = b^2$. I'm having problems determining if this relation is antisymmetric. I think it is. I did the following: Direct proof attempt (got stucked) We ...
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### discrete mathematics relations question 2

I am a little confused by this relation R3 is a subset of Z×Z defined by (x,y) in the set R3 if and only if x>2y is it reflexive? Symmetric? antisymmetric? or transitive? i say its NOT reflexive ...
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### Discrete mathematics Relations Question

if r2 is in the set of N*N ( natural numbers) with (X,y) in the subset of r2, if and only if x+y=0 is it reflexive? is it symmetric? is it anti symmetric? is it Transitive? i said it is reflective ...
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### Proving the symmetry of an equivalence relation

When proving the symmetry of an equivalence relation, must each equivalence class be closed under symmetry. for example: the relation both x and y > 10 or both x and y < 10 across all ...
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### Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
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### Monotonic Relation

Consider the relations $r_1, r_2, \ldots , r_n$ and consider a relational expression language as follows: $r_i$ is a relational expression, $1 \le i \le n$ $e_1 \circ e_2$ is a relational ...