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Recent questions in Discrete Mathematics
2
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181
GO Classes Set Theory And Algebra Practice Set 1 | Question: 5
Define $\mathcal{R}$ the binary relation on $\mathbb{N} \times \mathbb{N}$ to mean $(a, b) \mathcal{R}(c, d)$ iff $b \mid d$ and $a \mid c$ $\mathcal{R}$ is symmetric but not reflexive. $\mathcal{R}$ is transitive and symmetric but not reflexive $\mathcal{R}$ is reflexive and transitive but not symmetric None of the above
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182
GO Classes Set Theory And Algebra Practice Set 1 | Question: 6
Let $A$ be any set. Subset Relation on $\mathrm{P}(\mathrm{A})$ is Anti-symmetric?
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183
GO Classes Set Theory And Algebra Practice Set 1 | Question: 7
Define $\mathcal{R}$ the binary relation on $\mathbb{N} \times \mathbb{N}$ to mean $(a, b) \mathcal{R}(c, d)$ iff $b \mid d$ and $a \mid c$
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GO Classes Set Theory And Algebra Practice Set 1 | Question: 8
Consider the following binary relations on the naturals (non-negative integers). Which ones are reflexive? Symmetric? Anti-symmetric? Transitive? Partial orders? Justify your claims. $A(x, y)$, defined to be true if and only if $y$ ... because eight comes before eighty-one, and $E(8,8)$ is true because eight comes no later than eight.)
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185
GO Classes Set Theory And Algebra Practice Set 1 | Question: 9
The relation $\mathcal{R}$ on $\mathbb{Q}$ with $\forall x, y \in \mathbb{Q}: x \sim y$ if $x y=0$. The relation "has the same mother" on the set of students. The relation $\sim$ on $\mathbb{Q}$ where $a, b \in \mathbb{Q}$ ... $\{a, b, c\}$. The relation $\{(x, x),(y, y)\}$ on $\{x, y\}.$
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May 28, 2023
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186
GO Classes Set Theory And Algebra Practice Set 1 | Question: 10
Are the following relations reflexive, symmetric, transitive, antisymmetric? Explain. Let $R$ be a relation on $\mathbb{Z}$ such that $(a, b) \in R$ iff $b=a$ or $b=-a$. Let $R$ be a relation on $\mathbb{R}$ ... $(a, b) \in R$ iff $a+b$ is a rational number, that is can be represented by a fraction.
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May 28, 2023
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187
GO Classes Set Theory And Algebra Practice Set 1 | Question: 11
Let $A \neq \varnothing$ be a set. Consider the following statements: $\varnothing$ is a reflexive binary relation on $A$; $\varnothing$ is a symmetric binary relation on $A ;$ $\varnothing$ is a transitive binary relation on $A$; Which of ... $(2)$ are correct. Only $(2)$ and $(3)$ are correct. None is correct. All are correct.
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May 28, 2023
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188
GO Classes Set Theory And Algebra Practice Set 1 | Question: 12
Define the binary relation $R$ on the set $A:=\{-4,-3,-2,-1,1,2,3,4\}$ as follows: $ (x, y) \in R \Longleftrightarrow\left|x^2-y^2\right| \leqslant 5 $ for all $x, y \in A$. Which of the following statements ... at all. $R$ is reflexive. $R$ is irreflexive. $R$ is transitive. $R$ is symmetric. $R$ is asymmetric. $R$ is antisymmetric.
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189
GO Classes Set Theory And Algebra Practice Set 1 | Question: 13
$ R=\left\{(x, y) \in \mathbb{N}^2: \exists n \in \mathbb{N}, x^n=y\right\} $ is a binary relation on the set of natural numbers $\mathbb{N}$. Determine which of the following properties ... symmetric Transitive For each property, either justify that the property always holds or show by a counterexample that the property does not hold.
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190
GO Classes Set Theory And Algebra Practice Set 1 | Question: 14
Determine if each of the following relations is reflexive, symmetric, antisymmetric, or transitive. Indicate if the relation is an equivalence relation. $R_1=\{(a, b) \mid-1 \leq a-b \leq 1\}$ on $\mathbf{R}$ ... $\mathbf{N}$ $ R_7=\left\{(a, b) \mid \frac{a}{b} \in \mathbf{Z}\right\}$ on $\mathbf{Z}$
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May 28, 2023
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191
GO Classes Set Theory And Algebra Practice Set 1 | Question: 15
Let $R \subseteq \mathbb{N} \times \mathbb{N}$ be a relation ( $a$ binary relation) on the set of natural numbers defined as follows: $ (x, y) \in R \Leftrightarrow x+y \geq 18 \text {. } $ is $R$ reflexive ... . is R symmetric? Prove your answer. Is $R$ antisymmetric? Prove your answer. Is $\mathrm{R}$ transitive? Prove your answer.
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May 28, 2023
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192
GO Classes Set Theory And Algebra Practice Set 1 | Question: 16
Among reflexive, symmetric, antisymmetric, and transitive, which of those properties are true of the above relation? It is both reflexive and symmetric It is only reflexive It is only antisymmetric It is both reflexive and transitive
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193
GO Classes Set Theory And Algebra Practice Set 1 | Question: 17
Among reflexive, symmetric, antisymmetric, and transitive, which of those properties are true of the above relation? It is both symmetric and transitive It is both reflexive and transitive It is reflexive, antisymmetric, and transitive It is both reflexive and antisymmetric
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May 28, 2023
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194
GO Classes Set Theory And Algebra Practice Set 1 | Question: 18
Among reflexive, symmetric, antisymmetric, and transitive, which of those properties are true of the above relation? It is only reflexive It is reflexive, symmetric, and transitive It is both reflexive and antisymmetric It is both reflexive and symmetric
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May 28, 2023
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195
GO Classes Set Theory And Algebra Practice Set 1 | Question: 19
Among reflexive, symmetric, antisymmetric, and transitive, which of those properties are true of the above relation? It is only transitive It is both antisymmetric and transitive It is both reflexive and transitive It has none of those properties
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196
GO Classes Set Theory And Algebra Practice Set 1 | Question: 20
Let $R$ be the relation on $M=\{1,2,3\}$ with the following diagraph representation: Then $R$ is not reflexive, not symmetric, and not transitive $R$ is transitive but not reflexive $R$ is an equivalence relation $R$ is symmetric but not transitive $R$ is reflexive but not symmetric
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197
GO Classes Set Theory And Algebra Practice Set 1 | Question: 21
Define the relation $\mathrm{O}$ on $\mathrm{Z}$ as follows: $ \forall m, n \in Z, m O n \longleftrightarrow \exists k \in Z \mid(m-n)=2 k+1 $ ... $\mathrm{O}$ is not reflexive, symmetric, and not transitive.
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May 28, 2023
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198
GO Classes Set Theory And Algebra Practice Set 1 | Question: 22
Given the relation $R=\{(n, m)|n, m \in \mathbb{Z}| n,|\neq| m \mid\}$. Which of the following statements about $R$ is correct? $R$ is not an equivalence relation because it is not reflexive or ... relation because it is not antisymmetric $R$ is not an equivalence relation because it is not symmetric $R$ is an equivalence relation
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196
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GO Classes Set Theory And Algebra Practice Set 1 | Question: 23
Determine whether the following relations are reflexive, symmetric, antisymmetric, and/or transitive: The empty relation $\text{R}=\{\}$ is defined on the natural numbers. The complete relation $\mathrm{R}=\mathbf{N} \times \mathbf{N}$ defined on the ... $\mathrm{R}$ on the integers where $a\text{R}b$ means $a^2=b^2$.
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GO Classes Set Theory And Algebra Practice Set 1 | Question: 24
For each of the following relations $R$ on the set of real numbers, decide whether it is reflexive, symmetric, and/or transitive? Justify your arguments. Is the relation an equivalence relation? Explain. $(x, y) \in R$ if and only if $|x-y| \leq 3$ ... $(x, y) \in R$ if and only if $|x+y|=|x|+|y|$.
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