I have the following proposition $x,y\in \mathbb{R},\mathrm{\exists}x\mathrm{\exists}y({x}^{2}+{y}^{2}<2xy)$. Is this proposition true or false? How can get its negation using Morgan's law?

Frank Day
2022-07-08
Answered

Is this proposition true or false? What is the negation of the proposition?

I have the following proposition $x,y\in \mathbb{R},\mathrm{\exists}x\mathrm{\exists}y({x}^{2}+{y}^{2}<2xy)$. Is this proposition true or false? How can get its negation using Morgan's law?

I have the following proposition $x,y\in \mathbb{R},\mathrm{\exists}x\mathrm{\exists}y({x}^{2}+{y}^{2}<2xy)$. Is this proposition true or false? How can get its negation using Morgan's law?

You can still ask an expert for help

furniranizq

Answered 2022-07-09
Author has **20** answers

Explanation:

It is false since the negation ${\mathrm{\forall}}_{x}{\mathrm{\forall}}_{y}({x}^{2}+{y}^{2}-2xy=(x-y{)}^{2}\ge 0)$ is true.

It is false since the negation ${\mathrm{\forall}}_{x}{\mathrm{\forall}}_{y}({x}^{2}+{y}^{2}-2xy=(x-y{)}^{2}\ge 0)$ is true.

asked 2021-08-02

Suppose that A is the set of sophomores at your school and B is the set of students taking discrete mathematics at your school. Express each of these sets in terms of A and B.

a) the set of sophomores taking discrete mathematics in your school

b) the set of sophomores at your school who are not taking discrete mathematics

c) the set of students at your school who either are sophomores or are taking discrete mathematics

Use these symbols: $\cap \cup$

asked 2021-08-18

Discrete Mathematics Basics

1) Find out if the relation R is transitive, symmetric, antisymmetric, or reflexive on the set of all web pages.where $(a,b)\in R$ if and only if

I)Web page a has been accessed by everyone who has also accessed Web page b.

II) Both Web page a and Web page b lack any shared links.

III) Web pages a and b both have at least one shared link.

asked 2021-07-28

Let A, B, and C be sets. Show that

asked 2020-11-09

Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.

asked 2022-05-23

Prove that a relation isn't transitive

Let $\begin{array}{r}{M}_{R}=\left(\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right)\end{array}$

Where ${M}_{R}$ is the relation matrix for a relation R. Is R reflexive, symmetric, antisymmetric or transitive?

I find that is Symmetric but isn't reflexive and antisymmetric, To verify if ${M}_{R}$ is transitive. I compute the Boolean product

$\begin{array}{r}\left(\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right)\odot \left(\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right)=\left(\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)\end{array}$

That means that ${M}_{R}\odot {M}_{R}\ne {M}_{R}$, So ${M}_{R}$ isn't transitive. This is correct?

Let $\begin{array}{r}{M}_{R}=\left(\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right)\end{array}$

Where ${M}_{R}$ is the relation matrix for a relation R. Is R reflexive, symmetric, antisymmetric or transitive?

I find that is Symmetric but isn't reflexive and antisymmetric, To verify if ${M}_{R}$ is transitive. I compute the Boolean product

$\begin{array}{r}\left(\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right)\odot \left(\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right)=\left(\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)\end{array}$

That means that ${M}_{R}\odot {M}_{R}\ne {M}_{R}$, So ${M}_{R}$ isn't transitive. This is correct?

asked 2022-09-04

Discrete Math - Relations and Matrix Representations

Are these answers correct? Do we assume p is created from S twice? Binary relation p on the set $S=\{a,b,c,d,e\}$ is defined as: $\ufffcp=\{(a,c),(a,e),(b,a),(e,d)\}$. ￼

1. What is the matrix representation of p?

2. Is p a reflexive relation?

Please explain. ￼

(1.) Would the matrix representation of p be following: a 1 placed at the intersections of (a,c),(a,e),(b,a),(e,d) and the rest zeros where $a-e$ is listed for columns and rows?

(2.) p is not a reflexive relation because for every element a in A, there is not an ordered pair (a,a) in the relation.

Are these answers correct? Do we assume p is created from S twice? Binary relation p on the set $S=\{a,b,c,d,e\}$ is defined as: $\ufffcp=\{(a,c),(a,e),(b,a),(e,d)\}$. ￼

1. What is the matrix representation of p?

2. Is p a reflexive relation?

Please explain. ￼

(1.) Would the matrix representation of p be following: a 1 placed at the intersections of (a,c),(a,e),(b,a),(e,d) and the rest zeros where $a-e$ is listed for columns and rows?

(2.) p is not a reflexive relation because for every element a in A, there is not an ordered pair (a,a) in the relation.

asked 2022-04-10

Riordan numbers recurrence

Let be ${C}_{n}$ the ${n}^{th}$ Catalan's number. Well, I have the following relation:

$f(n)=\sum _{k=0}^{n}(-1{)}^{n-k}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}{C}_{k}\text{.}$

I would like to know, if there is a way to obtain the recurrence:

$f(n)=\frac{n-1}{n+1}(2f(n-1)+3f(n-2))$ just by the first identity.

Let be ${C}_{n}$ the ${n}^{th}$ Catalan's number. Well, I have the following relation:

$f(n)=\sum _{k=0}^{n}(-1{)}^{n-k}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}{C}_{k}\text{.}$

I would like to know, if there is a way to obtain the recurrence:

$f(n)=\frac{n-1}{n+1}(2f(n-1)+3f(n-2))$ just by the first identity.