Anokuhle Soganga
2022-10-02

You can still ask an expert for help

asked 2021-08-15

How many elements are in the set
{ 0, { { 0 } }?

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 2022-09-06

Concrete mathematics: Computing the value of certain infinite sums example

In Concrete Mathematics (Graham, Knuth, Patashnik), on page 58, there is the below example of calculating the value of an infinite sum:

$$\begin{array}{rl}\sum _{k\ge 0}\frac{1}{(k+1)(k+2)}& =\sum _{k\ge 0}{k}^{\underset{\_}{-2}}\\ & =\underset{n\to \mathrm{\infty}}{lim}\sum _{k=0}^{n}{k}^{-2}=\underset{n\to \mathrm{\infty}}{lim}\frac{{k}^{\underset{\_}{-1}}}{-1}{{\textstyle |}}_{0}^{n}=1\end{array}$$

For that last part I don't understand how it is 1 and not -1. To start with, at 0, the summation property gives us 0:

$$\frac{{0}^{\underset{\_}{-1}}}{-1}=\frac{\frac{0}{0+1}}{-1}=0$$

Then 1 is

$$\frac{{1}^{\underset{\_}{-1}}}{-1}=\frac{\frac{1}{1+1}}{-1}=-\frac{1}{2}$$

and 2 is

$$\frac{{2}^{\underset{\_}{-1}}}{-1}=\frac{\frac{1}{2+1}}{-1}=-\frac{1}{3}$$

And so on tending towards -1 for larger n. As I understand it (the subtraction vertical bar notation) if n is, say, 2 (a long way from infinity to be sure) then we'd get $-\frac{1}{3}-0=-\frac{1}{3}$. And so on getting closer to -1 as best I can tell.

In Concrete Mathematics (Graham, Knuth, Patashnik), on page 58, there is the below example of calculating the value of an infinite sum:

$$\begin{array}{rl}\sum _{k\ge 0}\frac{1}{(k+1)(k+2)}& =\sum _{k\ge 0}{k}^{\underset{\_}{-2}}\\ & =\underset{n\to \mathrm{\infty}}{lim}\sum _{k=0}^{n}{k}^{-2}=\underset{n\to \mathrm{\infty}}{lim}\frac{{k}^{\underset{\_}{-1}}}{-1}{{\textstyle |}}_{0}^{n}=1\end{array}$$

For that last part I don't understand how it is 1 and not -1. To start with, at 0, the summation property gives us 0:

$$\frac{{0}^{\underset{\_}{-1}}}{-1}=\frac{\frac{0}{0+1}}{-1}=0$$

Then 1 is

$$\frac{{1}^{\underset{\_}{-1}}}{-1}=\frac{\frac{1}{1+1}}{-1}=-\frac{1}{2}$$

and 2 is

$$\frac{{2}^{\underset{\_}{-1}}}{-1}=\frac{\frac{1}{2+1}}{-1}=-\frac{1}{3}$$

And so on tending towards -1 for larger n. As I understand it (the subtraction vertical bar notation) if n is, say, 2 (a long way from infinity to be sure) then we'd get $-\frac{1}{3}-0=-\frac{1}{3}$. And so on getting closer to -1 as best I can tell.

asked 2022-06-16

How many strings of five decimal digits must be starting or ending with an odd number?

How many strings of five decimal digits must be starting or ending with an odd number?

Everywhere, I looked over the internet they used this method:

Thus, number of ways $=5\cdot 10\cdot 10\cdot 10\cdot 10=50000$ ways.

Thus, number of ways $=10\cdot 10\cdot 10\cdot 10\cdot 5=50000$ ways.

Therefore, the strings of five decimal digits that start with an odd number or end with an odd number $=50000+50000=100000$.

How many strings of five decimal digits must be starting or ending with an odd number?

Everywhere, I looked over the internet they used this method:

Thus, number of ways $=5\cdot 10\cdot 10\cdot 10\cdot 10=50000$ ways.

Thus, number of ways $=10\cdot 10\cdot 10\cdot 10\cdot 5=50000$ ways.

Therefore, the strings of five decimal digits that start with an odd number or end with an odd number $=50000+50000=100000$.

asked 2022-08-31