Recent questions in Advanced Math

Upper Level MathOpen question

orrocksydnie33 2022-09-22

find the 52nd term of the arithmetic sequence -8,-12,-16

Discrete mathAnswered question

Sneha Loganathan2022-09-16

Let f be the function from R to R defined by f(x)=x^2.Find f^-1({x|0<x<1})

Upper Level MathOpen question

rennieb2018 2022-09-11

what is the difference between a System of equations and an Equation?

Discrete mathAnswered question

Kiana Arias 2022-09-07

Question: Provide further motivation for defining $p\to q$ to be true when p is false For the first change, we call the resulting operator imp1.

Show that p imp1 q logically equivalent q imp1 p.

$$\begin{array}{|lll|}\hline p& q& p\phantom{\rule{thickmathspace}{0ex}}imp1\phantom{\rule{thickmathspace}{0ex}}\\ T& T& T\\ T& F& F\\ F& T& F\\ F& F& T\\ \hline\end{array}$$

Show that p imp1 q logically equivalent q imp1 p.

$$\begin{array}{|lll|}\hline p& q& p\phantom{\rule{thickmathspace}{0ex}}imp1\phantom{\rule{thickmathspace}{0ex}}\\ T& T& T\\ T& F& F\\ F& T& F\\ F& F& T\\ \hline\end{array}$$

Discrete mathAnswered question

Frida Faulkner 2022-09-07

Expressing statements in Discrete math

Given that

A is the set of all Alpha's

M is the set of all Men

how do I express this statement: Not all Alpha's are Men

.............

My attempt:

$A\subset S=0$

in other words saying that A is not a subset of S, but I can't use the not subset symbol on this problem.

Given that

A is the set of all Alpha's

M is the set of all Men

how do I express this statement: Not all Alpha's are Men

.............

My attempt:

$A\subset S=0$

in other words saying that A is not a subset of S, but I can't use the not subset symbol on this problem.

Discrete mathAnswered question

Lucille Douglas 2022-09-07

Question about getting a formula for a recurrence relations

So basically, I was watching video above which is on recurrence relations and I had a question about this statement:

$${a}_{n}={a}_{n-1}+6{a}_{n-2}$$

I understand how he got the $(-2{)}^{n}$ and $(3{)}^{n}$, but not about how he is adding them and then multiplying them by the variables $\alpha $ and $\beta $. He said that there is a proof online about why there is always going to be an $\alpha $ and a $\beta $ that will always make this statement true, but I wasn't able to find it and I was hoping that somebody could give me a step by step explanation about how this is. I was also wondering about the general case for getting a formula for recurrence relations in this form.

Note: I read somewhere that you can derive this from generating functions, but I don't have a strong background in them, so I was wondering if there is another way to derive the relation.

So basically, I was watching video above which is on recurrence relations and I had a question about this statement:

$${a}_{n}={a}_{n-1}+6{a}_{n-2}$$

I understand how he got the $(-2{)}^{n}$ and $(3{)}^{n}$, but not about how he is adding them and then multiplying them by the variables $\alpha $ and $\beta $. He said that there is a proof online about why there is always going to be an $\alpha $ and a $\beta $ that will always make this statement true, but I wasn't able to find it and I was hoping that somebody could give me a step by step explanation about how this is. I was also wondering about the general case for getting a formula for recurrence relations in this form.

Note: I read somewhere that you can derive this from generating functions, but I don't have a strong background in them, so I was wondering if there is another way to derive the relation.

Discrete mathAnswered question

Paul Reilly 2022-09-07

Need help understanding $\mathrm{\exists}x\mathrm{\forall}yvs\mathrm{\forall}x\mathrm{\exists}y$

My understanding is that for $\mathrm{\exists}x\mathrm{\forall}y$, there can only be one x value that is true for every single y value. Meaning theres only one x value (which cannot be changed) for every single different y value. The statement $\mathrm{\exists}x\mathrm{\forall}y(p(x,y))$ is true when there is one x value (lets say $x=0$) that is true for $y=-2,-1,0,1,2$,... (for every single y). Correct me if I am wrong but this is my understanding of this notation.

And now my understanding for the second notation $\mathrm{\forall}x\mathrm{\exists}y(p(x,y))$ is that for every x value, there exists a y such that p(x,y). Meaning for every x value ($x=-2,-1,0,1,2,...$) there can be a different y value for each x value so that the statement is true.

I dont really know how to explain this well but I'll try to summarize my understanding. If the notation is $\mathrm{\exists}x\mathrm{\forall}y$ then theres only one x that cannot be changed that is true for every y. If the notation is $\mathrm{\forall}x\mathrm{\exists}y$ then the y value doesnt have to be the same y value for every x value. Meaning for every x value there can be a y value that is different than another y value for another x value.

My understanding is that for $\mathrm{\exists}x\mathrm{\forall}y$, there can only be one x value that is true for every single y value. Meaning theres only one x value (which cannot be changed) for every single different y value. The statement $\mathrm{\exists}x\mathrm{\forall}y(p(x,y))$ is true when there is one x value (lets say $x=0$) that is true for $y=-2,-1,0,1,2$,... (for every single y). Correct me if I am wrong but this is my understanding of this notation.

And now my understanding for the second notation $\mathrm{\forall}x\mathrm{\exists}y(p(x,y))$ is that for every x value, there exists a y such that p(x,y). Meaning for every x value ($x=-2,-1,0,1,2,...$) there can be a different y value for each x value so that the statement is true.

I dont really know how to explain this well but I'll try to summarize my understanding. If the notation is $\mathrm{\exists}x\mathrm{\forall}y$ then theres only one x that cannot be changed that is true for every y. If the notation is $\mathrm{\forall}x\mathrm{\exists}y$ then the y value doesnt have to be the same y value for every x value. Meaning for every x value there can be a y value that is different than another y value for another x value.

Discrete mathAnswered question

Jadon Stein 2022-09-07

Discrete Math: Combinatorics and recursion

3. Let S be a set of size 37, and let x, y, and z be three distinct elements of S. How many subsets of S are there that contain x and y, but do not contain z?

(a) ${2}^{33}$

(b) ${2}^{34}$

(c) ${2}^{35}$

(d) ${2}^{37}-{2}^{35}-{2}^{36}$

(d) none of the above

Why is it B) I thought there is size 37 so it is 37 - 2 Is it because there is size 37 and for x and y; you do 37-2. but you cannot have z so you minus another 1. so $37-2-1=34$; ${2}^{34}$

12. The Fibonacci numbers are defined as follows: $f0=0,f1=1$, and $fn=fn-1+fn-2$ for $n\ge 2$. Consider again the recursive algorithm Fib, which takes as input an integer $n\ge 0$:

Algorithm Fib(n):

if $n=0\text{}or\text{}n=1$

then $f=n$

else $f=Fib(n-1)+Fib(n-2)$

end if;

return f

For $n\ge 3$, run algorithm Fib(n) and let an be the number of times that Fib(2) is called. Which of the following is true?

(a) For $n\ge 3$, ${a}_{n}={f}_{n-1}$

(b) For $n\ge 3$, ${a}_{n}={f}_{n}$

(c) For $n\ge 3$, ${a}_{n}={f}_{n+1}$

(d) For $n\ge 3$, ${a}_{n}=-1+{f}_{n}$

So if it is $$n=3$$ i will call fib(2) 1 time and if $n=4$ then fib(2) is called 2 times

How do I put this into an equation like above?

3. Let S be a set of size 37, and let x, y, and z be three distinct elements of S. How many subsets of S are there that contain x and y, but do not contain z?

(a) ${2}^{33}$

(b) ${2}^{34}$

(c) ${2}^{35}$

(d) ${2}^{37}-{2}^{35}-{2}^{36}$

(d) none of the above

Why is it B) I thought there is size 37 so it is 37 - 2 Is it because there is size 37 and for x and y; you do 37-2. but you cannot have z so you minus another 1. so $37-2-1=34$; ${2}^{34}$

12. The Fibonacci numbers are defined as follows: $f0=0,f1=1$, and $fn=fn-1+fn-2$ for $n\ge 2$. Consider again the recursive algorithm Fib, which takes as input an integer $n\ge 0$:

Algorithm Fib(n):

if $n=0\text{}or\text{}n=1$

then $f=n$

else $f=Fib(n-1)+Fib(n-2)$

end if;

return f

For $n\ge 3$, run algorithm Fib(n) and let an be the number of times that Fib(2) is called. Which of the following is true?

(a) For $n\ge 3$, ${a}_{n}={f}_{n-1}$

(b) For $n\ge 3$, ${a}_{n}={f}_{n}$

(c) For $n\ge 3$, ${a}_{n}={f}_{n+1}$

(d) For $n\ge 3$, ${a}_{n}=-1+{f}_{n}$

So if it is $$n=3$$ i will call fib(2) 1 time and if $n=4$ then fib(2) is called 2 times

How do I put this into an equation like above?

Discrete mathAnswered question

kybudmanqm 2022-09-07

Let G be a 3-regular plane graph with 12 faces. How many vertices does G have?

This would be pretty easy to solve if I knew that G is connected by using Eulers formula $|V|-|E|+|F|=2$.

But I don't know how to show that G is connected. Am I on the wrong path? Or is there some combinatorial argument to count the vertices?

This would be pretty easy to solve if I knew that G is connected by using Eulers formula $|V|-|E|+|F|=2$.

But I don't know how to show that G is connected. Am I on the wrong path? Or is there some combinatorial argument to count the vertices?

Discrete mathAnswered question

calcific5z 2022-09-07

Coefficient of expansion discrete math

What is the coefficient of ${x}^{12}{y}^{12}$ in the expansion of $(3x-7y{)}^{24}$? I am just checking if I answered it in the correct way. Since its the expansion to the power of 24 and $(xy{)}^{12}=({x}^{12})({y}^{12})$ then i Just substituted 3x and -7y with x and y. I got $(3x{)}^{12}(-7y{)}^{12}{\textstyle (}\genfrac{}{}{0ex}{}{24}{12}{\textstyle )}$. Is my reasoning correct or is there much more to this.

What is the coefficient of ${x}^{12}{y}^{12}$ in the expansion of $(3x-7y{)}^{24}$? I am just checking if I answered it in the correct way. Since its the expansion to the power of 24 and $(xy{)}^{12}=({x}^{12})({y}^{12})$ then i Just substituted 3x and -7y with x and y. I got $(3x{)}^{12}(-7y{)}^{12}{\textstyle (}\genfrac{}{}{0ex}{}{24}{12}{\textstyle )}$. Is my reasoning correct or is there much more to this.

Discrete mathAnswered question

Makayla Reilly 2022-09-07

Let $h=g\circ f\circ g$ where $f:\mathbb{R}\to \mathbb{Z}$ is the floor function and $g:\mathbb{R}\to \mathbb{R}:x\mapsto -x$.

(i) Compute h(3.4), h(7) and h(-1.3).

(ii) Describe what h is doing to a general real number x.

(i) Compute h(3.4), h(7) and h(-1.3).

(ii) Describe what h is doing to a general real number x.

Discrete mathAnswered question

driliwra7 2022-09-07

I wanted to ask here if this proving process were correct:

$f(a)=$ a div d

We must show this is an onto function

If it is onto, then $\mathrm{\forall}y$ in the codomain, $\mathrm{\exists}a$ in the domain such that $f(a)=y$

Consider an arbitrary y in the codomain. We know that $f(a)=y$ if and only if a div d $=y$.

a div $d=y$ implies that $a=dy+r$ .

But then $f(dy+r)=y$ because:

$f(dy+r)$= $(dy+r)$ div d meanining that

$dy+r=dq+k$ where q=$(dy+r)$ div d

Now since $a=dy+r$, a mod d is equal to $(dy+r)$ mod d so $k=r$

This means that:

$dy+r=dq+r$, so the quotient q of $dy+r$ is y. But then $f(dy+r)=y$, so for an arbitray y, there is an element $dy+r$ in the domain such that $f(dy+r)$ is y, as we wanted to show.

$f(a)=$ a div d

We must show this is an onto function

If it is onto, then $\mathrm{\forall}y$ in the codomain, $\mathrm{\exists}a$ in the domain such that $f(a)=y$

Consider an arbitrary y in the codomain. We know that $f(a)=y$ if and only if a div d $=y$.

a div $d=y$ implies that $a=dy+r$ .

But then $f(dy+r)=y$ because:

$f(dy+r)$= $(dy+r)$ div d meanining that

$dy+r=dq+k$ where q=$(dy+r)$ div d

Now since $a=dy+r$, a mod d is equal to $(dy+r)$ mod d so $k=r$

This means that:

$dy+r=dq+r$, so the quotient q of $dy+r$ is y. But then $f(dy+r)=y$, so for an arbitray y, there is an element $dy+r$ in the domain such that $f(dy+r)$ is y, as we wanted to show.

Discrete mathAnswered question

cjortiz141t 2022-09-07

Relations between ordered pairs.

I am completely confused about this question, everytime I look back onto it I have a different idea on how to interpret it. Any help is appreciated.

A relation R is defined on ${\mathbb{Q}}^{2}$ by (a,b)R(c,d) if and only if there exists a real number $x\ge 1$ such that $a=dx$ and $c=bx$.

I need to show what type of relation this is, e.g. is it reflexive, transitive, symmetric....? Right now, I am just having a lot of trouble on how to interpret this and how to actually come up with a way of proving this.

I am completely confused about this question, everytime I look back onto it I have a different idea on how to interpret it. Any help is appreciated.

A relation R is defined on ${\mathbb{Q}}^{2}$ by (a,b)R(c,d) if and only if there exists a real number $x\ge 1$ such that $a=dx$ and $c=bx$.

I need to show what type of relation this is, e.g. is it reflexive, transitive, symmetric....? Right now, I am just having a lot of trouble on how to interpret this and how to actually come up with a way of proving this.

Discrete mathAnswered question

calcific5z 2022-09-07

Discrete Math (Combination with repetition)

The employee is distributing 7 objects among 4 containrs (Xi's).

Assuming the containers are $X1+X2+X3+X4=7$ where $Xi>=0\mathrm{\forall}1<=i<=4$.

Determine all integer solutions.

The employee is distributing 7 objects among 4 containrs (Xi's).

Assuming the containers are $X1+X2+X3+X4=7$ where $Xi>=0\mathrm{\forall}1<=i<=4$.

Determine all integer solutions.

Discrete mathAnswered question

nar6jetaime86 2022-09-07

Binary strings and discrete math

Let S be the set of binary strings of length 30 with 10 1’s and 20 0’s. Let A be the set of the first 30 positive integers {1,2,3,…,30}. Let B be the set of all subsets of A containing 10 numbers. Find a one-to-one correspondence between S and B.

Let S be the set of binary strings of length 30 with 10 1’s and 20 0’s. Let A be the set of the first 30 positive integers {1,2,3,…,30}. Let B be the set of all subsets of A containing 10 numbers. Find a one-to-one correspondence between S and B.

Upper Level MathAnswered question

sincsenekdq 2022-09-07

Find Upper Bound for $T(n)=T(n-1)+T(\frac{n}{2})+n$ with recursive tree method.

Discrete mathAnswered question

Kaleigh Ayers 2022-09-07

How to choose witnesses for asymptotic growth?

Prove:

$$4{n}^{5}\u201350{n}^{2}+10n\in \mathrm{\Theta}({n}^{5})$$

$$0\le {c}_{1}({n}^{5})\le 4{n}^{5}\u201350{n}^{2}+10n\le {c}_{2}({n}^{5})$$

$$0\le {c}_{1}\le 4\u201350/{n}^{3}+10/{n}^{4}\le {c}_{2}$$

Now, I know for the middle portion, so long as $n\ge 3$ then it will approach 4, leaving me with

$$0\le {c}_{1}\le 4\le {c}_{2}$$

However, I don't understand how to get my exact witnesses. Everything I found on the internet, people choose their witnesses but don't really explain how. Can I just arbitrarily choose anything between 0 and 4 for ${c}_{1}$ and anything above 4 for ${c}_{2}$?

Prove:

$$4{n}^{5}\u201350{n}^{2}+10n\in \mathrm{\Theta}({n}^{5})$$

$$0\le {c}_{1}({n}^{5})\le 4{n}^{5}\u201350{n}^{2}+10n\le {c}_{2}({n}^{5})$$

$$0\le {c}_{1}\le 4\u201350/{n}^{3}+10/{n}^{4}\le {c}_{2}$$

Now, I know for the middle portion, so long as $n\ge 3$ then it will approach 4, leaving me with

$$0\le {c}_{1}\le 4\le {c}_{2}$$

However, I don't understand how to get my exact witnesses. Everything I found on the internet, people choose their witnesses but don't really explain how. Can I just arbitrarily choose anything between 0 and 4 for ${c}_{1}$ and anything above 4 for ${c}_{2}$?

Students pursuing advanced Math are constantly dealing with advanced Math equations that are mostly used in space engineering, programming, and construction of AI-based solutions that we can see daily as we are turning to automation that helps us to find the answers to our challenges. If it sounds overly complex with subjects like exponential growth and decay, don’t let advanced math problems frighten you because these must be approached through the lens of advanced Math questions and answers. Regardless if you are dealing with simple equations or more complex ones, just break things down into several chunks as it will help you to find the answers.