# Questions tagged [relations]

This tag is intended for questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric…), composition of relations and similar stuff. More-or-less the things about relations taught in the first elementary set theory or discrete math course.

3,279 questions
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### If a relation is neither Symmetric nor Anti-Symmetric, can it still be an order of some kind?

Say I have a relation that is Irreflexive and Transitive, but neither Symmetric nor Anti-Symmetric, can it still be a strict partial and / or strict total ordering? I realise this is an edge case, I ...
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### Showing a relation in NxN is an equivalence relation, N denotes a set of positive integers

Let $N∈Z^+$ and P represents a relation in$N x N$defined by $(a,b)P(c,d)$ iff $a + d = b + c$ we have to show that P is an equivalence relation I tried to prove the reflexive property , then ...
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### relations, injective functions and proof of total ordering

I have recently started learning about injective functions and can understand them to a basic level. injective functions essentially equate to ...
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### Please help me about Relation… [on hold]

I'm learning about Relation now, and I've usually solved the same type of problem (2,2). I was going to solve this type of problem for the first time, but I don't know how to solve the problem, as the ...
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### Correctly understanding relations

I get that something such as R={(0, 0),(0, 1),(1, 1),(1, 2),(0, 2),(2, 2)} on the set {0, 1, 2} would be reflexive, anti-...
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### Is denseness a antisymmetric relation?

Upon discovering that denseness is transitive, I wondered if denseness is a partial ordering ($\iff$ reflexive, antisymmetric, transitive). To be more precise: Let $X$ be a topological space. Then ...
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### One one function set S

Consider the set $A ={1,2,3,4,5, \cdots ,n} .$ Let $S$ be the set of all one one function f from $A$ to $A$, such that $|f(1)-1|=|f(2)-2|=|f(3)-3|=.....=|f(n)-n|$ I need to find number of elements ...
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### To check that if this inequality is an equivalence relation on $\mathbb{Z}$ [on hold]

I proved this inequality ; Which is a relation on $\mathbb{Z}$ s.t a and b belongs to $\mathbb{Z}$ $$a^2 - b^2 \le 7$$ is reflexive , I'm stuck at the symmetry of this relation, can anyone help? ...
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### Alternative method for counting equivalence relations on {1,2,3,4}

So my method goes like this: We have 16 ordered pairs. If R is: Reflexive: It has to include $(1,1), (2,2), (3,3), (4,4)$. So $2$ choices for each of the remaining $12$ pairs. Symmetric: If it ...
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### Equivalence classes of a relation

Define on R the relation $xTy$ if and only if $cos^2(x) + sin^2 (y) = 1$. Prove that this is an equivalence relation and find R/T About that second part, what do the equivalence classes look like? I ...
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### Predicate logic deciding whether atomic formulae hold in interpretations

Consider the formula $\varphi$ of First-order logic defined as $\forall x\forall y((B(x,y) \land B(y,x)) \rightarrow (A(x)\land C(y)))$ State whether it holds in the following interpretations: ...
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### How can a relation be both irreflexive and antisymmetric?

Summary Irreflexivity occurs where nothing is related to itself. Anti-symmetry provides that whenever 2 elements are related "in both directions" it is because they are equal. These two concepts ...
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### S = { (a,b) be elements of Real Numbers| |a-b|<2 or |a-b|=2}. List three elements in S?

I know or at least correct me if I'm wrong two elements are {3,2}. How would I come to find another element?
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### How you call the relationship between variables $X$ and $Y$ if $X=1-Y$

How you call the relationship between two variables where one is equal to 1 minus the other. For example, if my variables are $X$ and $Y$ and I have that $X=1-Y$ I what to say the $X$ is the ________ ...
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### Question regarding relations [closed]

Here is the problem I got while i was reading a book known as Mordern Algebra. Prove that relation $R$ are in the set of integers I defined by $aRb$, if $a$ and $b$ are both odd, is symmetric and ...
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### Question involving proving equivalence relations

In order to prove if a relation is an equivalence relation, it needs to be show that is all of: Reflexive Symmetric Transitive Whilst I am familiar with this, I am unsure how to approach the ...
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### on the set of all integer,For all $a, b ∈ Z, a R b,$ $⇔ a | b,$ is R antisymmetric?

on the set of all integer,For all $a, b ∈ Z, a R b,$ $⇔ a | b,$ is R antisymmetric? the answer is symmetric but i dont know how to prove it and how to find the counter example $a,b \in Z$ $ka=b$...
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### Equivalence relation for homogeneous coordinates

My geometry textbook states that the vectors $(a, b, c)^T$ and $k(a, b, c)^T$ represent the same line for any non-zero $k$; in other words, two such vectors related by an overall scaling are ...
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### Suppose $(S, ∼)$ is an equivalence relation and suppose $a, b ∈ S$. Show $[a] = [b]$ if $a ∼ b$ and $[a] ∩ [b] = ∅$ if $a \not\sim b$.

I am a bit lost on this question to the point that I don't know where to start. I am confused as to how I am supposed to show this without a defined ~ relation. any help would be greatly appreciated, ...
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### Counting the number of equivalence classes given the relation (a,b),(c,d) elements of AXA where (a,b)R(c,d) if and only if a+b=c+d

So i have a discrete math final and I don't know why but the profs decided not to post answer key for this one final and I need to check my understanding of this question. Let \$A = \{1, 2, 3, 4, \...