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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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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 $f(Z)$ $f(N)$ $f(R)$
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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 $S= \left \{ 2,1,0,1,2,3 \right \}$ $S= \left \{ 0,1,2,3,4,5 \right \}$ $S= \left \{ 1,5,7,11 \right \}$ $S= \left \{ 2,6,,10,14 \right \}$
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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 $f(x) =1$ $f(x) = 2x+1$ $f(x) =\left \lceil x/5 \right \rceil$ $f(x) = \left \lfloor (x^2+1)/3 \right \rfloor$
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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.
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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
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8
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
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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
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10
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.
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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.
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Kenneth Rosen Edition 7th Exercise 2.3 Question 23 (Page No. 153)
Determine whether each of these functions is a bijection from$R$ to $R.$ $f(x) = 2x+1$ $f(x) = x^2+1$ $f(x) = x^3$ $f(x) = (x^2+1)/(x^2+2)$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 22 (Page No. 153)
Determine whether each of these functions is a bijection from$R$ to $R.$ $f(x) = 3x+4$ $f(x) = 3x^2+7$ $f(x) = (x+1)/(x+2)$ $f(x) = x^5+1$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 21 (Page No. 153)
Give an explicit formula for a function from the set of integers to the set of positive integers that is onetoone, but not onto. onto, but not onetoone. onetoone and onto. neither onetoone nor onto.
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15
Kenneth Rosen Edition 7th Exercise 2.3 Question 20 (Page No. 153)
Give an example of a function from $N$ to $N$ that is onetoone but not onto. onto but not onetoone. both onto and onetoone (but different from the identity function). neither onetoone nor onto.
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16
Kenneth Rosen Edition 7th Exercise 2.3 Question 17 (Page No. 153)
Consider these functions from the set of teachers in a school. Under what conditions is the function onetoone if it assigns to a teacher his or her office. assigned bus to chaperone in a group of buses taking students on a field trip. salary. social security number
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17
Kenneth Rosen Edition 7th Exercise 2.3 Question 16 (Page No. 153)
Consider these functions from the set of students in a discrete mathematics class. Under what conditions is the function onetoone if it assigns to a student his or her mobile phone number. student identification number. final grade in the class. home town.
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18
Kenneth Rosen Edition 7th Exercise 2.3 Question 15 (Page No. 153)
Determine whether the function $f: Z \times Z \rightarrow Z$ is onto if $f(m,n) = m+n$ $f(m,n) = m^2+n^2.$ $f(m,n) = m.$ $f(m,n) = n.$ $f(m,n) = mn.$
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19
Kenneth Rosen Edition 7th Exercise 2.3 Question 14 (Page No. 153)
Determine whether $f: Z \times Z \rightarrow Z$ is onto if $f(m,n) = 2mn.$ $f(m,n) = m^2n^2.$ $f(m,n) = m+n+1.$ $f(m,n) = mn.$ $f(m,n) = m^24.$
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20
Kenneth Rosen Edition 7th Exercise 2.3 Question 11 (Page No. 153)
Determine whether each of these functions form $[a,b,c,d]$ to itself is onto? $f(a)=b, f(b)=a,f(c)=c,f(d)=d$ $f(a)=b, f(b)=b,f(c)=d,f(d)=c$ $f(a)=d, f(b)=b,f(c)=c,f(d)=d$
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21
Kenneth Rosen Edition 7th Exercise 2.3 Question 13 (Page No. 153)
Determine whether each of these functions from $Z$ to $Z$ is onto?? $f(n) = n1$ $f(n) =n^2+1$ $f(n)= n^3$ $f(n) =\left \lceil n/2 \right \rceil$
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22
Kenneth Rosen Edition 7th Exercise 2.3 Question 12 (Page No. 153)
Determine whether each of these functions from $Z$ to $Z$ is onetoone. $f(n) = n1$ $f(n) =n^2+1$ $f(n)= n^3$ $f(n) =\left \lceil n/2 \right \rceil$
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4
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23
Kenneth Rosen Edition 7th Exercise 2.3 Question 10 (Page No. 153)
Determine whether each of these functions form $[a,b,c,d]$ to itself is onetoone. $f(a)=b, f(b)=a,f(c)=c,f(d)=d$ $f(a)=b, f(b)=b,f(c)=d,f(d)=c$ $f(a)=d, f(b)=b,f(c)=c,f(d)=d$
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24
Kenneth Rosen Edition 7th Exercise 2.3 Question 9 (Page No. 153)
Find the values. $\left \lceil 3/4 \right \rceil$ $\left \lfloor 7/8 \right \rfloor$ $\left \lceil 3/4 \right \rceil$ $\left \lfloor 7/8 \right \rfloor$ $\left \lceil 3 \right \rceil$ ... $\left \lfloor 1/2.\left \lfloor 5/2 \right \rfloor \right \rfloor$
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25
Kenneth Rosen Edition 7th Exercise 2.3 Question 8 (Page No. 153)
Find the values. $\left \lfloor1.1 \right \rfloor$ $\left \lceil 1.1 \right \rceil$ $\left \lfloor 0.1 \right \rfloor$ $\left \lceil 0.1 \right \rceil$ $\left \lceil 2.99 \right \rceil$ ... $\left \lceil \left \lfloor 1/2 \right \rfloor + \left \lceil 1/2 \right \rceil + 1/2 \right \rceil$
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Kenneth Rosen Edition 7th Exercise 2.3 Question 1 (Page No. 153)
Find the domain and range of these functions. the function that assigns to each pair of positive integers the maximum of these two integers the function that assigns to each positive integer the number of the digits ... of the first $1$ in the string and that assigns the value 0 to a bit string consisting of all 0s
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27
Kenneth Rosen Edition 7th Exercise 2.3 Question 6 (Page No. 152)
Find the domain and range of these functions. the function that assigns to each pair of positive integers the first integer of the pair the function that assigns to each positive integer its largest decimal digit the function ... root of the integer the function that assigns to a bit string the longest string of ones in the string
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28
Kenneth Rosen Edition 7th Exercise 2.3 Question 5 (Page No. 152)
Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. the function that assigns to each bit string the number of ... of 8 bits) the function that assigns to each positive integer the largest perfect square not exceeding this integer
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29
Kenneth Rosen Edition 7th Exercise 2.3 Question 4 (Page No. 152)
Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. the function that assigns to each nonnegative integer its last digit the ... of one bits in the string the function that assigns to a bit string the number of bits in the string
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30
Kenneth Rosen Edition 7th Exercise 2.3 Question 3 (Page No. 152)
Determine whether $f$ is a function from the set of all bit strings to the set of integers if $f(S)$is the position of a $0$ bit in $S$. $f(S)$is the number of $1$ bits in $S$. $f(S)$is the smallest integer $i$ such that the $i$ th bit of $S$ is $1$ and $f(S)=0$ when $S$ is the empty string, the string with no bits.
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