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1

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.

Define the relation $\mathrm{O}$ on $\mathrm{Z}$ as follows:$$\forall m, n \in Z, m O n \longleftrightarrow \exists k \in Z \mid(m-n)=2 k+1$$Which one of the following st...

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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.

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 $b=-a$.Le...

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3

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.

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 $...

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4

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 $|x+y|=|x|+|y|$.

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 ...

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5

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

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 ...

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6

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.)

Consider the following binary relations on the naturals (non-negative integers). Which ones are reflexive? Symmetric? Anti-symmetric? Transitive? Partial orders? Justify ...

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7

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 $\mathbf{Z}$

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...

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8

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.

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 $x, y \in A$...

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9

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.

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 \g...

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10

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.

$$R=\left\{(x, y) \in \mathbb{N}^2: \exists n \in \mathbb{N}, x^n=y\right\}$$is a binary relation on the set of natural numbers $\mathbb{N}$.Determine which of the follow...

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11

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 $a \mid c$

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: 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 $\{x, y\}.$

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 ...

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13

GO Classes Set Theory And Algebra Practice Set 1 | Question: 4
Let $A$ be the set of all ordered pairs of integers, that is, $A=Z \times Z$. Define a binary relation $R$ on $A$ as follows: for all $(a, b),(c, d) \in A$ ... reflexive? Is $R$ symmetric? Is $R$ antisymmetric? Is $R$ transitive? Is $R$ an equivalence relation, a partial order, neither, or both?

Let $A$ be the set of all ordered pairs of integers, that is, $A=Z \times Z$. Define a binary relation $R$ on $A$ as follows: for all $(a, b),(c, d) \in A$,$$(a, b) R(c, ...

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14

GO Classes Set Theory And Algebra Practice Set 1 | Question: 3
The following table describes a binary relation. Find the set of ordered pairs that is this relation, as in the definition of a binary relation. ...

The following table describes a binary relation. Find the set of ordered pairs that is this relation, as in the definition of a binary relation.$$\begin{array}{c|c|c|c|c|...

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15

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

Among reflexive, symmetric, antisymmetric, and transitive, which of those properties are true of the above relation?It is both symmetric and transitiveIt is both reflexiv...

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16

GO Classes Set Theory And Algebra Practice Set 1 | Question: 1
Let $A=\{0,1,2,3\}$ and $R$ a relation over $A$ : $ R=\{(0,0),(0,1),(0,3),(1,1),(1,0),(2,3),(3,3)\} $ Draw the directed graph of $R$. Check whether $R$ is an equivalence relation. Give a counterexample in each case in which the relation does not satisfy one of the properties of being an equivalence relation.

Let $A=\{0,1,2,3\}$ and $R$ a relation over $A$ :$$R=\{(0,0),(0,1),(0,3),(1,1),(1,0),(2,3),(3,3)\}$$Draw the directed graph of $R$. Check whether $R$ is an equivalence re...

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17

GO Classes Set Theory And Algebra Practice Set 1 | Question: 2
$\begin{array}{l|llllll} \textbf{Relations on}\; \mathbb{Z}: & \quad < & \qquad \leq & \qquad = & \qquad \mid & \qquad \nmid & \qquad \neq \\ \hline \hline \text{Reflexive} & \\ \text{Symmetric} & \\ \text{Transitive} & \end{array}$ Fill "Yes or No" in the above table.

$$\begin{array}{l|llllll}\textbf{Relations on}\; \mathbb{Z}: & \quad < & \qquad \leq & \qquad = & \qquad \mid & \qquad \nmid & \qquad \neq \\\hline \hline \text{Reflexive...

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18

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

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 r...

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19

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

Among reflexive, symmetric, antisymmetric, and transitive, which of those properties are true of the above relation?It is both reflexive and symmetricIt is only reflexive...

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20

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

Among reflexive, symmetric, antisymmetric, and transitive, which of those properties are true of the above relation?It is only transitiveIt is both antisymmetric and tran...

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