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Kenneth Rosen Edition 7th Exercise 2.3 Question 65 (Page No. 155)
Draw the graph of the function $f(n) =$\left \lfloor x \right \rfloor +\left \lfloor x/2 \right \rfloor$ from $R$ to $R$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 64 (Page No. 155)
Draw the graph of the function $f(n) =$\left \lfloor x/2 \right \rfloor$ from $R$ to $R$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 63 (Page No. 155)
Draw the graph of the function $f(n) =$\left \lfloor 2x \right \rfloor$ from $R$ to $R$
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4
Kenneth Rosen Edition 7th Exercise 2.3 Question 62 (Page No. 155)
Draw the graph of the function $f(n) = 1n^2$ from $Z$ to $Z$
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5
Kenneth Rosen Edition 7th Exercise 2.3 Question 61 (Page No. 155)
Data are transmitted over a particular Ethernet network in blocks of $1500$ octets (blocks of $8$ bits). How many blocks are required to transmit the following amounts of data over this Ethernet network? (Note that a byte is a synonym ... $1.544$ $\text{megabytes}$ of data $45.3$ $\text{megabytes of}$ data
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Kenneth Rosen Edition 7th Exercise 2.3 Question 60 (Page No. 155)
How many ATM cells (described in Example 28) can be transmitted in $10$ seconds over a link operating at the following rates? $128$ kilobits per second ($1$ kilobit= $1000$ bits) $300$ kilobits per second $1$ megabit per second ($1$ megabit=$1,000,000$ bits)
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7
Kenneth Rosen Edition 7th Exercise 2.3 Question 59 (Page No. 155)
How many bytes are required to encode $n$ bits of data where $n$ equals $7$ $17$ $1001$ $28800$
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8
Kenneth Rosen Edition 7th Exercise 2.3 Question 58 (Page No. 154)
How many bytes are required to encode $n$ bits of data where $n$ equals $4$ $10$ $500$ $3000$
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9
Kenneth Rosen Edition 7th Exercise 2.3 Question 57 (Page No. 154)
Let $a$ and $b$ be real numbers with $a<b$. Use the floor and / or ceiling functions to express the number of integers $n$ that satisfy the inequality $a<n<b.$
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10
Kenneth Rosen Edition 7th Exercise 2.3 Question 56 (Page No. 154)
Let $a$ and $b$ be real numbers with $a<b$. Use the floor and / or ceiling functions to express the number of integers $n$ that satisfy the inequality $a≤n≤b$.
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Apr 11, 2019
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Set Theory & Algebra
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11
Kenneth Rosen Edition 7th Exercise 2.3 Question 55 (Page No. 154)
The function INT is found on some calculators, where INT$(x)$ = $\left \lfloor x \right \rfloor$ when $x$ nonnegative real number and INT$(x)$ = $\left \lceil x \right \rceil$ when x is a negative real number. Show that this INT function satisfies the identity INT$(x)$=$$ INT$(x)$
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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$\left \lceil x \right \rceil = \left \lfloor x \right \rfloor$
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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.
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Apr 9, 2019
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14
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 $
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15
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 $\left \lceil x+m \right \rceil = \left \lceil x \right \rceil +m.$
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16
Kenneth Rosen Edition 7th Exercise 2.3 Question 49 (Page No. 154)
Show that if $x$ is a real number, then $x1 < \left \lfloor x \right \rfloor <= x<= \left \lceil x \right \rceil < x+1.$
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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.
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18
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
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19
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
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20
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 $f^{1}\sim(S) = \sim f^{1}(S).$
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21
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 $f^{1}(S \cup T) = f^{1}(S) \cup f^{1}(T)$ $f^{1}(S \cap T) = f^{1}(S) \cap f^{1}(T)$
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22
Kenneth Rosen Edition 7th Exercise 2.3 Question 43 (Page No. 154)
Let $g(x) = \left \lfloor x \right \rfloor$. Find $g^{1}(\left \{ 0 \right \})$ $g^{1}(\left \{ 1,0,1 \right \})$ $g^{1}(\left \{ x0<x<1 \right \})$
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23
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 $f^{1}(\left \{ 1 \right \})$ $f^{1}(\left \{ x0<x<1 \right \})$ $f^{1}(\left \{ xx>4 \right \})$
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24
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 $f(S \cup T) = f(S) \cup f(T)$ $f(S \cap T) = f(S) \cap f(T)$
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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 $f$.
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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 $fog =gof$
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27
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 $R$.
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28
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.
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29
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.
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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.
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