# Tagged Questions

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### Relations examples (reflexivity, symmetry, transitivity)

I've found the two textbooks I'm using to to be particularly unhelpful in explaining these concepts, especially as they relate to English examples (non-existent). The first few following questions ...
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### Properties of a relation on matrices: $(m_1,m_2)\in R$ iff $m_1\cdot m_2$ is defined

Let $M$ be a set of matrices of integers. Let $R$ be the relation on $M$ defined as follows: For any two $m_1, m_2 \in M, (m_1, m_2) \in R$ iff the matrix multiplication $m_1 \cdot m_2$ is defined. ...
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### Why relations are defined as the smallest

Often relations are defined as follows: The xxxxx relation is the smallest relation satisfying... My question is why relations are defined as the smallest ...
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### If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
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### Binary relation, reflexive, symmetric and transitive

I have a question regarding an image. I'm currently studying binary relations and the following image confused me: What got me confused is that the page from which I got the link ...
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### Need help proving that $fRg \Leftrightarrow fg = f$ on $B^{n}$ to $B$ if and only if $f(b_1,…,b_n) \leq g(b_1,…,b_n)$

I'm trying to gather my thoughts for proving the following claim: For $fRg \Leftrightarrow fg = f$ on $B^{n}$ to $B$, show that $fRg$ if and only if for any input values $b_1,...,b_n$, we ...
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### Equivalence Relation ~

let S = {1,2,3,4} Explain why each of the below are not equivalence relation. { (1,1), (1,2), (2,1), (2,2), (3,3) } { (1,1), (1,2), (2,3), (1,3), (2,2), (3,3), (4,4) } { (1,1), (2,2), (3,3), ...
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### Partition induced by a Relation

Here's the problem: Let $A=\{1,2,3,4,5,6,7,8,9\}$. Define a relation $R$ on set $A$ by $xRy$ if and only if $2\mid(x+y)$ Assuming that $R$ is an equivalence relation, determine the partition of set ...
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### Need help clarifying relation properties

So I am facing some issues determining the right properties for: $xRy\;if\,\sin^2(x) + \cos^2(y) = 1$. (On real numbers) Obviously this one is reflexive as $\sin^2(x) + \cos^2(x) = 1$ is a basic ...
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### Bijection of a function.

Define the function f: $(2,\infty) -> (-\infty,-1)$ by $f(x)= \frac{-x}{x-2}$. Show that f is bijective. I know i need to prove both injective and surjective, and I was able to solve the equation ...
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### Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive

I am just looking for some clarification on this exercise: Let $A = \{a,b,c,d\}$. Give an example of a relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive. I understand that if I ...
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### Symmetric relation , why are these symmetric?

$R_1 = \{(a,b)$ such that $a \leq b \}$ $R_2 = \{(a,b)$ such that $a>b \}$ $R_3 = \{(a,b)$ such that $a=b$ or $a=-b \}$ $R_4 = \{(a,b)$ such that $a=b \}$ $R_5 = \{(a,b)$ such that $a=1+b \}$ ...
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### Is a relation induced by a partition always an equivalence relation?

Is a relation induced by a partition always an equivalence relation? I'm having some serious trouble understanding this concept and I was wondering if this is true.
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### Proving a relation is a total order relation

Consider question #21 part a: Here is the solution: However, consider the definition of a total order relation: The solution didn't prove that the relation is a partial order relation. This ...
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### How to count the number of distinct equivalence classes for a relation involving truth tables?

I am having trouble with question 22 part (2): Here is the solution: How did the author know that there are 256 distinct equivalence classes? Where did they get $2^8$ from?
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### Equivalence Relation on R (real numbers)

Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z. (a) Is R an equivalence relation? Prove your answer. ...
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### Let A = {1,2,3,4} Let F be the set of all functions from A to A. (check the parts)

Let $\operatorname{S}$ be a relation on $F$ defined by: $\forall f, g \in F, f\,\operatorname{S}\,g \iff f(i) = g(i), \exists i \in A$. (a) Recall that the identity function $I_A : A \mapsto A$ is ...
### Equivalence relation question with cardinality and countability $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q$
Let $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q$ What is the cardinality of $[\pi]_S$ ? Prove that the quotient group $\mathbb R/S$ is uncountable. Well I think that cardinality is ...