please answer question 2, and 4 2. (al Determine all bijections from the HI, 2,

Question

please answer question 2, and 4

2. (al Determine all bijections from the HI, 2, 3I into ka, b, cl. b Determine all bijections from II, 2, 3I into la, b, c, dl. 3. Which of the following are one-toone, onto, orboth? R defined by fi(x) -x. a) f :R :Z Z defined by 2. N defined by f j, k) 2/3. (d) fa :P P defined by fa (n) n/21, where xisthe ceiling of t, the smallest integer greater than or equal to r. N defined by n (f) f :N Nx N defined by fo (n) 2n, 2n+ 4. Which of the following are injections sujections,orbijections on R,the setofreal numbers? X20

Explanation / Answer

2.)

a.)

bijections from {1,2,3} into {a,b,c}

{(1,a),(2,b),(3,c)},{(1,a),(2,c),(3,b)},{(1,b),(2,a),(3,c)},{(1,c),(2,a),(3.b)},{(1,c),(2,b),(3,a)},{(1,b),(2,c),(3,a)}

b.)

bijections from {1,2,3} into {a,b,c,d}

don't exist because cardinalities of the two sets are not equal

4.

a.)

f(x) = -2x is bijection

for all y in R, f(x) = y has a solution x in R

b.)

g(x) = x^2 - 1 is not a bijection

for y<-1, g(x) = y has no solution x in R

c.)

h(x) ={x^2 if x>=0, x if x<0} is a bijection

for all y in R, h(x) = y has a solution x in R

d.)

q(x)=2^x is not a bijection

for y<0, q(x) = y has no solution x in R

e.)

r(x)=x^3 is a bijection

for all y in R, f(x) = y has a solution x in R

f.)

s(x)=x^3-x is not a bijection

s(x)=0 has two solutions x = -1 and x = 1

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