Login
Register
Dark Mode
Brightness
Profile
Edit Profile
Messages
My favorites
My Updates
Logout
Recent questions tagged discrete-mathematics
0
votes
0
answers
661
Kenneth Rosen Edition 7 Exercise 2.5 Question 17 (Page No. 176)
If $A$ is an uncountable set and $B$ is a countable set, must $A − B$ be uncountable?
If $A$ is an uncountable set and $B$ is a countable set, must $A − B$ be uncountable?
admin
257
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
662
Kenneth Rosen Edition 7 Exercise 2.5 Question 16 (Page No. 176)
Show that a subset of a countable set is also countable.
Show that a subset of a countable set is also countable.
admin
185
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
663
Kenneth Rosen Edition 7 Exercise 2.5 Question 15 (Page No. 176)
Show that if $A$ and $B$ are sets, $A$ is uncountable, and $A \subseteq B,$ then $B$ is uncountable
Show that if $A$ and $B$ are sets, $A$ is uncountable, and $A \subseteq B,$ then $B$ is uncountable
admin
207
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
664
Kenneth Rosen Edition 7 Exercise 2.5 Question 14 (Page No. 176)
Show that if $A$ and $B$ are sets with the same cardinality, then $\mid A \mid \leq \mid B \mid $ and $\mid B \mid \leq \mid A\mid.$
Show that if $A$ and $B$ are sets with the same cardinality, then $\mid A \mid \leq \mid B \mid $ and $\mid B \mid \leq \mid A\mid.$
admin
194
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
665
Kenneth Rosen Edition 7 Exercise 2.5 Question 13 (Page No. 176)
Explain why the set $A$ is countable if and only if $\mid A \mid \leq \mid Z^{+}\mid.$
Explain why the set $A$ is countable if and only if $\mid A \mid \leq \mid Z^{+}\mid.$
admin
182
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
666
Kenneth Rosen Edition 7 Exercise 2.5 Question 12 (Page No. 176)
Show that if $A$ and $B$ are sets and $A \subset B$ then $\mid A \mid \leq \mid B\mid.$
Show that if $A$ and $B$ are sets and $A \subset B$ then $\mid A \mid \leq \mid B\mid.$
admin
175
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
667
Kenneth Rosen Edition 7 Exercise 2.5 Question 11 (Page No. 176)
Give an example of two uncountable sets $A$ and $B$ such that $A \cap B$ is finite. countably infinite. uncountable
Give an example of two uncountable sets $A$ and $B$ such that $A \cap B$ isfinite.countably infinite.uncountable
admin
185
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
668
Kenneth Rosen Edition 7 Exercise 2.5 Question 10 (Page No. 176)
Give an example of two uncountable sets $A$ and $B$ such that $A − B$ is finite. countably infinite. uncountable.
Give an example of two uncountable sets $A$ and $B$ such that $A − B$ isfinite.countably infinite.uncountable.
admin
180
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
669
Kenneth Rosen Edition 7 Exercise 2.5 Question 9 (Page No. 176)
Suppose that a countably infinite number of buses, each containing a countably infinite number of guests, arrive at Hilbert’s fully occupied Grand Hotel. Show that all the arriving guests can be accommodated without evicting any current guest.
Suppose that a countably infinite number of buses, each containing a countably infinite number of guests, arrive at Hilbert’s fully occupied Grand Hotel. Show that all ...
admin
188
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
670
Kenneth Rosen Edition 7 Exercise 2.5 Question 8 (Page No. 176)
Show that a countably infinite number of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
Show that a countably infinite number of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
admin
220
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
671
Kenneth Rosen Edition 7 Exercise 2.5 Question 7 (Page No. 176)
Suppose that Hilbert’s Grand Hotel is fully occupied on the day the hotel expands to a second building which also contains a countably infinite number of rooms. Show that the current guests can be spread out to fill every room of the two buildings of the hotel.
Suppose that Hilbert’s Grand Hotel is fully occupied on the day the hotel expands to a second building which also contains a countably infinite number of rooms. Show th...
admin
209
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
672
Kenneth Rosen Edition 7 Exercise 2.5 Question 6 (Page No. 176)
Suppose that Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel.
Suppose that Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel.
admin
212
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
1
votes
0
answers
673
Kenneth Rosen Edition 7 Exercise 2.5 Question 5 (Page No. 176)
Show that a finite group of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
Show that a finite group of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
admin
207
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
674
Kenneth Rosen Edition 7 Exercise 2.5 Question 4 (Page No. 176)
Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. integers not divisible by $3$ integers divisible ... of all $1s$ the real numbers with decimal representations of all $1s\: \text{or}\: 9s$
Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive i...
admin
369
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
675
Kenneth Rosen Edition 7 Exercise 2.5 Question 3 (Page No. 176)
Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. all bit strings not ... in their decimal representation the real numbers containing only a finite number of $1s$ in their decimal representation
Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive i...
admin
374
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
1
answer
676
Kenneth Rosen Edition 7 Exercise 2.5 Question 2 (Page No. 176)
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. the integers greater than $10$ the odd negative ... $A = \{2, 3\}$ the integers that are multiples of $10$
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the...
admin
2.8k
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
1
answer
677
Kenneth Rosen Edition 7 Exercise 2.5 Question 1 (Page No. 176)
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. the negative integers the even ... $1,000,000,000$ the integers that are multiples of $7$
Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the...
admin
1.3k
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
678
Kenneth Rosen Edition 7 Exercise 2.4 Question 46 (Page No. 170)
Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we specify that $0! = 1.$ $\displaystyle{}\prod_{i=0}^{4} j\:!$
Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we s...
admin
218
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
679
Kenneth Rosen Edition 7 Exercise 2.4 Question 45 (Page No. 170)
Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we specify that $0! = 1.$ $\displaystyle{}\sum_{i=0}^{4} j\:!$
Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we s...
admin
171
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
680
Kenneth Rosen Edition 7 Exercise 2.4 Question 44 (Page No. 170)
Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we specify that $0! = 1.$ Express $n!$ using product notation.
Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we s...
admin
179
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
681
Kenneth Rosen Edition 7 Exercise 2.4 Question 43 (Page No. 170)
What are the values of the following products? $\displaystyle{}\prod_{i=0}^{10} i$ $\displaystyle{}\prod_{i=5}^{8} i$ $\displaystyle{}\prod_{i=1}^{100} (-1)^{i}$ $\displaystyle{}\prod_{i=1}^{10} 2$
What are the values of the following products?$\displaystyle{}\prod_{i=0}^{10} i$$\displaystyle{}\prod_{i=5}^{8} i$$\displaystyle{}\prod_{i=1}^{100} (-1)^{i}$$\displaysty...
admin
166
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
682
Kenneth Rosen Edition 7 Exercise 2.4 Question 42 (Page No. 169)
Find a formula for $\displaystyle{}\sum_{k=0}^{m}\left \lfloor \sqrt{k} \right \rfloor ,$ when $m$ is a positive integer. There is also a special notation for products. The product of $a_{m}, a_{m+1},\dots,a_{n}$ ... $j = m\: \text{to}\: j = n\: \text{of}\: a_{j} .$
Find a formula for $\displaystyle{}\sum_{k=0}^{m}\left \lfloor \sqrt{k} \right \rfloor ,$ when $m$ is a positive integer.There is also a special notation for products. T...
admin
214
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
683
Kenneth Rosen Edition 7 Exercise 2.4 Question 41 (Page No. 169)
Find a formula for $\displaystyle{}\sum_{k=0}^{m}\left \lfloor \sqrt{k} \right \rfloor ,$ when $m$ is a positive integer.
Find a formula for $\displaystyle{}\sum_{k=0}^{m}\left \lfloor \sqrt{k} \right \rfloor ,$ when $m$ is a positive integer.
admin
191
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
684
Kenneth Rosen Edition 7 Exercise 2.4 Question 39 (Page No. 169)
Find $\displaystyle{}\sum_{k=100}^{200}k. \text{(Use Table 2.)}$
Find $\displaystyle{}\sum_{k=100}^{200}k. \text{(Use Table 2.)}$
admin
136
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
685
Kenneth Rosen Edition 7 Exercise 2.4 Question 38 (Page No. 169)
Use the technique given in question $35,$ together with the result of question $37b,$ to derive the formula for $\displaystyle{}\sum_{k=1}^{n}k^{2}$ given in Table $2.\:\:[$Hint: Take $a_{k} = k^{3}$ in the telescoping sum in question $35.]$
Use the technique given in question $35,$ together with the result of question $37b,$ to derive the formula for $\displaystyle{}\sum_{k=1}^{n}k^{2}$ given in Table $2.\:\...
admin
175
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
686
Kenneth Rosen Edition 7 Exercise 2.4 Question 37 (Page No. 169)
Sum both sides of the identity $k^{2}-(k-1)^{2} = 2k-1$ from $k=1$ to $k=n$ and use question $35$ to find a formula for $\displaystyle{}\sum_{k = 1}^{n}(2k − 1)$ (the sum of the first $n$ odd natural numbers). a formula for $\displaystyle{}\sum_{k = 1}^{n} k.$
Sum both sides of the identity $k^{2}-(k-1)^{2} = 2k-1$ from $k=1$ to $k=n$ and use question $35$ to finda formula for $\displaystyle{}\sum_{k = 1}^{n}(2k − 1)$ (the su...
admin
237
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
687
Kenneth Rosen Edition 7 Exercise 2.4 Question 36 (Page No. 169)
Version$1:$ Use the identity $\dfrac{1}{k(k+1)} = \dfrac{\left(\frac{1}{k-1}\right)}{(k+1)}$ and question $35$ to compute $\displaystyle{}\sum_{k=1}^{n} \dfrac{1}{k(k+1)}.$ Version$2:$ Use the identity $1/(k(k + 1)) = 1/k − 1/(k + 1)$ and question $35$ to compute $\displaystyle{}\sum_{k=1}^{n} 1/(k(k+1)).$
Version$1:$ Use the identity $\dfrac{1}{k(k+1)} = \dfrac{\left(\frac{1}{k-1}\right)}{(k+1)}$ and question $35$ to compute $\displaystyle{}\sum_{k=1}^{n} \dfrac{1}{k(k+1)}...
admin
132
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
688
Kenneth Rosen Edition 7 Exercise 2.4 Question 35 (Page No. 169)
Show that $\displaystyle{}\sum_{j=1}^{n}(a_{j} - a_{j-1}) = a_{n} -a_{0,}$ where $a_{0}, a_{1},\dots,a_{n}$ is a sequence of real numbers. This type of sum is called telescoping.
Show that $\displaystyle{}\sum_{j=1}^{n}(a_{j} - a_{j-1}) = a_{n} -a_{0,}$ where $a_{0}, a_{1},\dots,a_{n}$ is a sequence of real numbers. This type of sum is called tele...
admin
140
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
689
Kenneth Rosen Edition 7 Exercise 2.4 Question 34 (Page No. 169)
Compute each of these double sums. $\displaystyle{}\sum_{i=1}^{3}\sum_{j=1}^{2}(i-j)$ $\displaystyle{}\sum_{i=0}^{3}\sum_{j=0}^{2}(3i+2j)$ $\displaystyle{}\sum_{i=1}^{3}\sum_{j=0}^{2}j$ $\displaystyle{}\sum_{i=0}^{2}\sum_{j=0}^{3}i^{2}j^{3}$
Compute each of these double sums.$\displaystyle{}\sum_{i=1}^{3}\sum_{j=1}^{2}(i-j)$$\displaystyle{}\sum_{i=0}^{3}\sum_{j=0}^{2}(3i+2j)$$\displaystyle{}\sum_{i=1}^{3}\sum...
admin
171
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
0
votes
0
answers
690
Kenneth Rosen Edition 7 Exercise 2.4 Question 33 (Page No. 169)
Compute each of these double sums. $\displaystyle{}\sum_{i=1}^{2}\sum_{j=1}^{3}(i+j)$ $\displaystyle{}\sum_{i=0}^{2}\sum_{j=0}^{3}(2i+3j)$ $\displaystyle{}\sum_{i=1}^{3}\sum_{j=0}^{2}i$ $\displaystyle{}\sum_{i=0}^{2}\sum_{j=1}^{3}ij$
Compute each of these double sums.$\displaystyle{}\sum_{i=1}^{2}\sum_{j=1}^{3}(i+j)$$\displaystyle{}\sum_{i=0}^{2}\sum_{j=0}^{3}(2i+3j)$$\displaystyle{}\sum_{i=1}^{3}\sum...
admin
188
views
admin
asked
Apr 21, 2020
Set Theory & Algebra
kenneth-rosen
discrete-mathematics
set-theory&algebra
descriptive
+
–
Page:
« prev
1
...
18
19
20
21
22
23
24
25
26
27
28
...
80
next »
Email or Username
Show
Hide
Password
I forgot my password
Remember
Log in
Register