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Recent questions tagged kennethrosen
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Kenneth Rosen Edition 7th Exercise 2.3 Question 54 (Page No. 154)
Prove that if $x$ is a reall number , then $\left \lfloor x \right \rfloor =  \left \lceil x \right \rceil$ and$\left \lceil x \right \rceil = \left \lfloor x \right \rfloor$
asked
Apr 11
in
Set Theory & Algebra
by
Pooja Khatri
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(
10.9k
points)

19
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kennethrosen
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2
Kenneth Rosen Edition 7th Exercise 2.3 Question 53 (Page No. 154)
Prove that if $n$ is an integer, then $\left \lfloor n/2 \right \rfloor = n/2$ if $n$ is even and $(n1)/2$ if $n$ is odd.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

12
views
kennethrosen
discretemathematics
settheory&algebra
0
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0
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3
Kenneth Rosen Edition 7th Exercise 2.3 Question 52 (Page No. 154)
Show that if $x$ is a real number and $n$ is an integer, then $x<=n$ if and only if $\left \lceil x \right \rceil <=n$ $n<=x$ if and only if $ n<=\left \lfloor x \right \rfloor $
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

24
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
4
Kenneth Rosen Edition 7th Exercise 2.3 Question 51 (Page No. 154)
Show that if $x$ is a real number and $n$ is an integer, then $x<n$ if and only if $\left \lfloor x \right \rfloor < n$ $n<x$ if and only if $ n<=\left \lfloor x \right \rfloor $
asked
Apr 9
in
Mathematical Logic
by
Pooja Khatri
Boss
(
10.9k
points)

33
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
1
answer
5
Kenneth Rosen Edition 7th Exercise 2.3 Question 50 (Page No. 154)
Show that if $x$ is a real number, and $m$ is an integer, then $\left \lceil x+m \right \rceil = \left \lceil x \right \rceil +m.$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

26
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
6
Kenneth Rosen Edition 7th Exercise 2.3 Question 49 (Page No. 154)
Show that if $x$ is a real number, then $x1 < \left \lfloor x \right \rfloor <= x<= \left \lceil x \right \rceil < x+1.$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

6
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
7
Kenneth Rosen Edition 7th Exercise 2.3 Question 48 (Page No. 154)
Show that if $x$ is a real number, then $\left \lceil x \right \rceil  \left \lfloor x \right \rfloor =1$ if $x$ is not an integer and $\left \lceil x \right \rceil  \left \lfloor x \right \rfloor =0$ if $x$ is an integer.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

18
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
8
Kenneth Rosen Edition 7th Exercise 2.3 Question 47 Page No. 154)
Show that $\left \lceil x1/2 \right \rceil$ is the closest integer to the number $x$,except when $x$ is midway between two integers, when it is the smaller of these two integers
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

4
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
9
Kenneth Rosen Edition 7th Exercise 2.3 Question 46 (Page No. 154)
Show that $\left \lfloor x+1/2 \right \rfloor$ is the closest integer to the number $x$,except when $x$ is midway between two integers, when it is the larger of these two integers
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

15
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
10
Kenneth Rosen Edition 7th Exercise 2.3 Question 45 (Page No. 154)
Let $f$ be a function from $A$ to $B$. Let $S$ be a subset of $B$. Show that $f^{1}\sim(S) = \sim f^{1}(S).$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

15
views
kennethrosen
discretemathematics
settheory&algebra
+1
vote
0
answers
11
Kenneth Rosen Edition 7th Exercise 2.3 Question 44 (Page No. 154)
Let $f$ be a function from $A$ to $B$. Let $S$ and $T$ be subsets of $B$. Show that $f^{1}(S \cup T) = f^{1}(S) \cup f^{1}(T)$ $f^{1}(S \cap T) = f^{1}(S) \cap f^{1}(T)$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

15
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
12
Kenneth Rosen Edition 7th Exercise 2.3 Question 43 (Page No. 154)
Let $g(x) = \left \lfloor x \right \rfloor$. Find $g^{1}(\left \{ 0 \right \})$ $g^{1}(\left \{ 1,0,1 \right \})$ $g^{1}(\left \{ x0<x<1 \right \})$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

16
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
13
Kenneth Rosen Edition 7th Exercise 2.3 Question 42 (Page No. 154)
Let $f$ be the function from $R$ to $R$ defined by $f(x) = x^2$. Find $f^{1}(\left \{ 1 \right \})$ $f^{1}(\left \{ x0<x<1 \right \})$ $f^{1}(\left \{ xx>4 \right \})$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

13
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
14
Kenneth Rosen Edition 7th Exercise 2.3 Question 40 (Page No. 154)
Let $f$ be a function from the set $A$ to the set $B$. Let $S$ adn$T$ be subsets of $A$ .Show that $f(S \cup T) = f(S) \cup f(T)$ $f(S \cap T) = f(S) \cap f(T)$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

7
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
15
Kenneth Rosen Edition 7th Exercise 2.3 Question 39 (Page No. 154)
Show that the function $f(x)=ax+b$ from $R$ to $R$ is invertible, where $a$ and $b$are constants, with $a=0$, and find the inverse of $f$.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

16
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
16
Kenneth Rosen Edition 7th Exercise 2.3 Question 38 (Page No. 154)
Let $f(x) = ax+b$ and $g(x) =cx+d$, where a,b,c, and d ar constants. Determine neccessary and sufficient conditions on the constants a,b,c, and d so that $fog =gof$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

6
views
kennethrosen
discretemathematics
settheory&algebra
+1
vote
1
answer
17
Kenneth Rosen Edition 7th Exercise 2.3 Question 36 (Page No. 154)
Find $fog$ and $gof$. Where $f(x) = x^2+1$ and $g(x)= x+2$, are functions from $R$ to $R$.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

29
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
18
Kenneth Rosen Edition 7th Exercise 2.3 Question 35 (Page No. 154)
If $f$ and $fog$ are onto, does it follow that $g$ is onto?Justify your answer.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

18
views
kennethrosen
discretemathematics
settheory&algebra
difficult
0
votes
0
answers
19
Kenneth Rosen Edition 7th Exercise 2.3 Question 34 (Page No. 154)
If $f$ and $fog$ are onetoone, does it follow that $g$ is onetoone? Justify your answer.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

6
views
kennethrosen
discretemathematics
settheory&algebra
difficult
0
votes
0
answers
20
Kenneth Rosen Edition 7th Exercise 2.3 Question 33 (Page No. 154)
Suppose that $g$ is a function from $A$ to $B$ and $f$ is a function from $B$ to $C$. Show that if both $f$ and $g$ are onetoone functions,then $fog$ is also onetoone. Show that if both $f$ and $g$ are onto functions, then $fog$ is also onto.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

4
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
21
Kenneth Rosen Edition 7th Exercise 2.3 Question 32 Page No. 154)
Let $f(x) = 2x$ where the domain is the set of real numbers. What is $f(Z)$ $f(N)$ $f(R)$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

6
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
22
Kenneth Rosen Edition 7th Exercise 2.3 Question 31 (Page No. 154)
Let $f(x) = \left \lfloor x^2/3 \right \rfloor$. Find $f(S)$ if $S= \left \{ 2,1,0,1,2,3 \right \}$ $S= \left \{ 0,1,2,3,4,5 \right \}$ $S= \left \{ 1,5,7,11 \right \}$ $S= \left \{ 2,6,,10,14 \right \}$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

5
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
23
Kenneth Rosen Edition 7th Exercise 2.3 Question 30 (Page No. 154)
Let $S= \left \{ 1,0,2,4,7 \right \}$. Find $f(S)$ if $f(x) =1$ $f(x) = 2x+1$ $f(x) =\left \lceil x/5 \right \rceil$ $f(x) = \left \lfloor (x^2+1)/3 \right \rfloor$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

7
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
24
Kenneth Rosen Edition 7th Exercise 2.3 Question 29 (Page No. 154)
Show that the function $f(x)=x$ from the set of real numbers to the set of nonnegative real numbers is not invertible, but if the domain is restricted to the set of nonnegative real numbers, the resulting function is invertible.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

6
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
25
Kenneth Rosen Edition 7th Exercise 2.3 Question 28 (Page No. 153)
Show that the function $f(x)=e^x $ from the set of real numbers to the set of real numbers is not invertible, but if the codomain is restricted to the set of positive real numbers, the resulting function is invertible
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

7
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
1
answer
26
Kenneth Rosen Edition 7th Exercise 2.3 Question 27 (Page No. 153)
Prove that a strictly decreasing function from $R$ to itself is onetoone. Give an example of an decreasing function from $R$ to itself that is not onetoone
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

14
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
27
Kenneth Rosen Edition 7th Exercise 2.3 Question 26 (Page No. 153)
Prove that a strictly increasing function from $R$ to itself is onetoone. Give an example of an increasing function from $R$ to itself that is not onetoone
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

19
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
28
Kenneth Rosen Edition 7th Exercise 2.3 Question 25 (Page No. 153)
Let $f: R \rightarrow R$ and let $f(x) >0$ for all $x \epsilon R.$ Show that $f(x) $ is strictly decreasing if and only if the function $g(x) = 1/f(x)$ is strictly increasing.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

4
views
kennethrosen
discretemathematics
settheory&algebra
0
votes
0
answers
29
Kenneth Rosen Edition 7th Exercise 2.3 Question 24 (Page No. 153)
Let $f: R \rightarrow R$ and let $f(x) >0$ for all $x \epsilon R.$ Show that $f(x) $ is strictly increasing if and only if the function $g(x) = 1/f(x)$ is strictly decreasing.
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

19
views
kennethrosen
discretemathematics
settheory&algebra
+1
vote
1
answer
30
Kenneth Rosen Edition 7th Exercise 2.3 Question 23 (Page No. 153)
Determine whether each of these functions is a bijection from$R$ to $R.$ $f(x) = 2x+1$ $f(x) = x^2+1$ $f(x) = x^3$ $f(x) = (x^2+1)/(x^2+2)$
asked
Apr 9
in
Set Theory & Algebra
by
Pooja Khatri
Boss
(
10.9k
points)

24
views
kennethrosen
discretemathematics
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