# Questions tagged [relations]

For questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric, etc), a composition of relations, or anything else concerning a relation on a set.

3,702 questions
Filter by
Sorted by
Tagged with
33 views

### How do I prove this relation is an equivalence relation?

Given a set $A=RxR$, consider a relation R defined by $(a,b)R(c,d)$ iff $b=d$ How do I prove that R is an equivalence relation?
35 views

### Proving $xRy\iff xy^{-1}\in\ker(f)$ is equivalent relation, where $f:(G,.) \to (H,.)$ is homomorphism of groups $G$ and $H$.

I know for sure it is but I failed to prove how. I tried to use the homomorphism definition where $f(x*y^{-1}) = f(x)*f(y^{-1})$ but I didn't see a pattern.
21 views

### Why do we sum to prove the transitive property?

In the relation $x − y = 4m$, to prove the transitive property, we do the following: $$(x − y) + (y − z) = 4m + 4n = 4(m + n)$$ Why?
22 views

### Equivalence classes for a relation between integers

I have the following relation in $\mathbb Z$: $x \sim y$ iff $x-y$ is a multiple of $4$. How do I approach this?
29 views

### Suppose $[a], [b] \in \mathbb{Z}_n$ and $[a]\cdot[b] = $. Is it necessarily true that either $[a] = $ or $[b] = $?

Let $n \in \mathbb{N}$. Let $R$ be the equivalence relation $\equiv \pmod{n}$. Suppose $[a], [b] \in \mathbb{Z}_n$ and $[a]\cdot[b] = $. Is it necessarily true that either $[a] = $ or $[b] = $...
34 views

### A binary relation contained in its square

My colleague (I guess, investigating structure of specific semigroups) is looking for references about binary relations $R\subset X\times X$ such that $R\subset R\circ R$, that is for each $(v,u)\in R$...
27 views

### Use a reflexive and transitive closure to transform an antisymmetric and acyclic relation into a partially ordered set.

The relation $R=\{(x,S_1),(S_1,S_2),(S_2,S_3),(S_3,S_4),(S_4,y)\}$ is antisymmetric and acyclic but not transitive or reflexive. We know that any antisymmetric and acyclic relation can be turned into ...
14 views

### Showing that a relation is neither an equivalence relation nor a partial order

Say we have a relation $R$ on $\mathbb{Z} \times \mathbb{Z}$ such that $(a, b) R (c, d)$ if $a^2 + b^2 \leq c^2 + d^2$ So to prove that $R$ is not an equivalence relation we need to show that $R$ ...
25 views

### Show that $P$ is a partition of the corresponding set $A$ and define the equivalence relation induced by partition $P$.

$A = \mathbb N$ and $P = \{ \{ m \in \mathbb N: m \ \ \text{is a multiple of} \ \ 3\}, \{ m \in \mathbb N: m \ \ \text{is not a multiple of 3}\} \}$. How can I define the equivalence relation ...
17 views

### Union of equivalence relation, problem If R ∪ S is an equivalence relation on A, then R and S are equivalence relations on A.

Let A be a non-empty set and let R and S be relations defined on A. If R ∪ S is an equivalence relation on A, then R and S are equivalence relations on A. how can i prove it? i think it is not ...
33 views

### Equivalence relation proof, problem $x∼y \iff x^2 − 2x + 1 = y^2 + 4y + 4$

Determine if the following relationships are equivalent. If they are, determine the equivalence classes, give a set of indices and the quotient set. Sketch out the graph of each relationship (whether ...
46 views

### Are sets predicates?

In so many different instances we need to be able to construct sets of functions. The first axiom of set theory (at least in the order that I learned) says that $a\in b$ is only a proposition if a and ...
26 views

### Is $G=\{(x,y)\in \mathbb{Z}\times\mathbb{Z}:x+y<4\}$ irreflexive?

Let $G=\{(x,y)\in \mathbb{Z}\times\mathbb{Z}:x+y<4\}$ Is $G$ irreflexive? I know $<$ is an irreflexive relation. But how to show it? How do I do this? The Reflexive and Irreflexive property ...
35 views

### Is this Hasse diagram drawn correctly? [closed]

Is this Hasse diagram of the divisibility relation when $A = \{3,4,5,6,7,8,9,10,11,12\}$ correct? Thanks for help. Image
7 views

### How does a transitive extension differ from a transitive closure?

Quoting an example from C.L Liu's Discrete Mathematics: Let R be a binary relation on A. The transitive extension of R (let's denote it as $R_1$) is a binary relation on A such that $R_1$ contains R. ...
71 views

### Are the set of all convergent geometric series whose sum is a rational number is countable? [closed]

I tried this way: As the sum of convergent geometric series is $\frac{a}{1-r}$ and $-1<r<1$. Moreover sum is also a rational number. So $a$ and $r$ should be rational numbers. As rational ...
31 views

19 views

### How to find the pre-image for [-3,3] [closed]

question image Determine wether or not each of the following relations are functions. If a relation is a function, find its range. If it is not a function, give reasons. Find the inverse if it has ...
14 views

### Define a relation R on Z × N by (a, α)R(b, β) if and only if aβ = bα. Prove that R is a reflexive relation.

I'm a bit confused about how to prove that R is reflexive. By definition, R, a relation in a set S, is reflexive if and only if ∀x∈S, xRx. Since (a, α)R(b, β), we know that aβ = bα. Then to prove ...
28 views

### Find these relations on $\mathbb{N}$.

Give an example of a relation on $\mathbb{N}$ which is, reflexive, transitive but not symmetric transitive, symmetric but not reflexive reflexive, symmetric but not transitive anti-symmetric, ...
52 views

### Find $f(x,y)$ when $f(x)$ and $f(y)$ are known [closed]

I have a problem related to the combination of 2 relations. I know the relation between the diffusion coefficient and the temperature (say D(T)) and I know the relation between the diffusion ...
16 views

### How to find the pre-image of a relation given the intrrval

enter image description here The questions are in the link above, can someone please explain how to find the pre-image of the interval B[-2,2]?
59 views

36 views

### What is the inverse relation of R

Determine the inverse relation $R^{−1}$ for the relation $R = \{(x,y) : x + 4y \text{ is odd}\}$ defined on $\mathbb{N}$. does this mean that $R^{-1} = \{(y,x):y+4x \text{ is odd}\}$ ?
22 views

### How is it defined that $a_1 \times b_1 < a_2 \times b_1$ and that $a_1 \times b_3 < a_3 \times b_1$?

I am reading Topology by James Munkres and he defines the dictionary order relation as: Definition Suppose that $A$ and $B$ are both sets with order relations $<_A$ and $<_B$ respectively. ...
26 views

### Transitive Relationships

I am examining transitive relationships and understand the premise that if $x \rightarrow y \rightarrow z$, then the relation needs to contain $x \rightarrow z$ to be considered transitive. My ...
23 views

### Is a relation given by {(a,a), (b,b), (c,c), (a,b), (a,c)} on set {a,b,c} a partial order?

Is was a TRUE/FALSE question in a discrete math exam. I answered False because I believe the relation is neither transitive or not transitive so the definition for a partial order doesn't apply (...
23 views

### Calculating a seen height of an object from a scope

I have an object that is 0.6 meter in diameter that stand at 1000 meter away from me, I have a scope with magnification of X15-40. How can I calculate the size of the image that I will see through ...
83 views

### Algorithm for generating subsets of finite sets

Obviously there is no method to generate a sequence of subsets of $\mathbb N$ that, in analogue with diagonalizing elements in $\mathbb N^2$, gives an arbitrary number of arbitrary elements. But is ...
82 views

### Is this another definition for inverse relation?

We know that if $R\colon A\to B$, we can define the inverse relation as follows: $$R^{-1}=\{(y,x)\in B\times A\mid(x,y)\in R\}.$$ Now I want to know if this set, let's call it $R\:'^{-1}$, is the same ...
29 views

### Inductive closure of a relation?

I did not really know whether to ask this here or in MathOverflow. On the one hand, I have a maths degree and this is part of my PhD research on computer science, and I am pretty sure this is not a ...
44 views

41 views

### Prove the following statement are equivalent

Let $|A| = |B| = n$ and let $f : A \to B$ be everywhere defined function. Prove that the following three statements are equivalent. $f$ is one to one. $f$ is onto. $f$ is one-to-one correspondence (...
My friend asked me this question. 1) Are i) $y=x^3+1$ (or in general $y=f(x)$) & ii)$y-x^3=1$ (or in general $\phi(x,y) = c$) both same? He thinks 1st one is a function $f:x \to y$ and 2nd ...