5

Suppose Xis nonempty subset of R that is bonded above: Let Y = {Zx:x € X} Let & supX. (1) If a is an upper bound ofX, show Za is an upper bound of Y: (Hen...

Question

Suppose Xis nonempty subset of R that is bonded above: Let Y = {Zx:x € X} Let & supX. (1) If a is an upper bound ofX, show Za is an upper bound of Y: (Hence, this shows that Za is an upper bound of Y.) (2) Use the definition of the least upper bound to prove that Za supY .

Suppose Xis nonempty subset of R that is bonded above: Let Y = {Zx:x € X} Let & supX. (1) If a is an upper bound ofX, show Za is an upper bound of Y: (Hence, this shows that Za is an upper bound of Y.) (2) Use the definition of the least upper bound to prove that Za supY .



Answers

Show that every nonempty finite subset of a lattice has a least upper bound and a greatest lower bound.

The set of all X. Such that the absolute value of X -1 is less than two. Find the least upper bound and the greatest lower bound of this set. Let's go ahead and try to grab this on the number line. All right, the absolute value of X -1 is less than two. So let's put some numbers in here zero one, two, three, four. And let's go down negative on negative two. -3. And that should be enough. Now the absolute value of X -1. That means the difference between X and one. Of course in absolute value turner's but the difference between X and one. So one is basically going to be the center of our graph. The distance between X and one is less than two. So the distance between wherever exes and one is less than two. So the distance that we can move, the distance that X can be from one. Is that most two spaces away? So two spaces away here would be three but we've got to be strictly less than two spaces away. So we're not going to really quite get to three or we could be a distance of two or two spaces away from one in the other direction which would bring us down to negative one. Okay, but not including -1. So if we shade in between negative one and positive one, not including negative one. I'm sorry if we if we shade in between negative one and three, Not including -1. Not including three. Then the shaded portion. This shaded segment interval basically represents the set of all X where the distance between X and one is less than two. Okay, so any X in the screen interval, the distance between an X. In this green interval and one is less than two. Okay, If we're equal to two in this direction you'd be at three. So shading everything to the left. The three means the distance from X to one is less than two. Same thing in this direction. Now that you see uh the interval shaded on the number line, it's a lot easier. The least upper bound upper bounds will be any numbers greater than the set. The least of those upper bounds is three. The greatest lower balance. Okay, Lower than the set. The greatest of the numbers. The greatest of the lower balance will be negative one. Every number in this set is greater than Or equal to in this case greater than negative ones. The -1 is the greatest of the lower bills.

If possible, we want to find the least upper bound and the greatest lower bound of the set of all X satisfying E T D X is less than one now. E T D X equals one when X is zero, so if X is less than zero, member eat E T D X Will Equal one. If x equals zero. So GTX will be less than one when X is less than zero. If E T X equals one. When X0, each zero would be one than when X is less than zero, E T X will be less than one. So all X satisfying each of the X less than one are really all X satisfying X less than zero, not including zero, but everything to the left of zero less than zero. So you can see that zero is an upper bound to this set and will actually be the least upper bound. So the least upper bound will be zero. There are no lower bounds. This set keeps decreasing towards negative infinity without bound. So there are no lower bounds. There is no greatest lower bound

Yeah If possible, find the least upper bound and the greatest. Lower bound for the set of all X. Such that the absolute value of X -1 is greater than two. The absolute value of X -1 is greater than two. In other words, the distance between X and one Is greater than two. The distance between X and one is greater than two. Here's one. Where can X be if the distance between X and one is greater than two? Well X has to be more than two units away from one. So this is one, then X can be anything above three can equal three because effects was three would be exactly a distance up to from one. We want the distance between X and one to be greater than to not equal to two. So X can be greater than three, not including three. And likewise in the other direction, if the distance between X and one is greater than two, X can be more than two units away from one in the other direction. So anything to the left or less than -1. So the shaded blue region represent the values of X that are represented in this set. So you can see that this set does not have an upper bound, It is boundless uh in the positive direction, it is also boundless in the negative direction, it keeps going off to posit Anthony and negative infinity. So there are no upper balance. So there is no least upper bound, there are no lower bounds. So there is no greatest lower bound

The set of all X. Such that the log of X is less than one. Find the least upper bound and the greatest lower belt. Now log of E A log of E equals one. So if you want the log of X to be less than one, you need X to be less than he because to log a B equals one. So log actually be less than one when X is less than eight. Now we also have to remember that you cannot take the log of zero and you cannot take the log of a negative number. So if you want the natural log of X to be less than one, you know that X has to be less than a. So X has to be strictly less than E. If you want the log of X to be strictly less than one. But because you cannot take the log of zero or negative number, X must also be greater than zero. So X needs to be less than E to satisfy this, but X has to be greater than zero. So that you're not trying to do something you're not able to do, you can take a lot of a zero or a negative number. So X must stay greater than zero. So X is going to be between zero and E. Not including zero. Not including the so now it's easy to see that E will be the least upper bound and zero will be the greatest lower bound of this set


Similar Solved Questions

5 answers
Froblem #I of two linear differential equalions follovt Solve the initial value problem for bysict X -2y, x(0)5x - Y'Y(0) minimum of R, and M is the maximum Kof R mM where m is the @igurea and Then calculate the numerical value numerical result for the value of a t two Round-off your Here is the value ol provide below (42 points):
Froblem #I of two linear differential equalions follovt Solve the initial value problem for bysict X -2y, x(0) 5x - Y' Y(0) minimum of R, and M is the maximum Kof R mM where m is the @igurea and Then calculate the numerical value numerical result for the value of a t two Round-off your Here is ...
5 answers
Problem 3 Solve the initial value problem by the method of Laplace transform. (If this problem is solved by different method sorry NO CREDIT:) y" + 4y' + 3y = 4, y(o) 9 (0)
Problem 3 Solve the initial value problem by the method of Laplace transform. (If this problem is solved by different method sorry NO CREDIT:) y" + 4y' + 3y = 4, y(o) 9 (0)...
5 answers
At 5"C, the ion-product constant of water; Kw, is 1.87 x 10^-15. The pH of pure water at 5'C is:Oz.ooo 7.464 6.784 7.364 None of the above is correct:
At 5"C, the ion-product constant of water; Kw, is 1.87 x 10^-15. The pH of pure water at 5'C is: Oz.ooo 7.464 6.784 7.364 None of the above is correct:...
5 answers
1 Exercise. Uclug t dcfiultion provided 1 Theelopr u{be7 }LEnb}
1 Exercise. Uclug t dcfiultion provided 1 Theelopr u{be 7 } LEnb}...
5 answers
[0/5 Points]DETAILSPREVIOUS ANSWERSHOLTLINAFind all values of h such that the vectors {a1 a2, a3} span R? where a2 -[BH
[0/5 Points] DETAILS PREVIOUS ANSWERS HOLTLINA Find all values of h such that the vectors {a1 a2, a3} span R? where a2 -[BH...
5 answers
Find the limit: Use /'Hospital's Rule if appropriate. Use INF to represent positive infinity, NINF for negative infinity, and D for the limit does not exist. 2ez 2 21 lim x-+0 4x2
Find the limit: Use /'Hospital's Rule if appropriate. Use INF to represent positive infinity, NINF for negative infinity, and D for the limit does not exist. 2ez 2 21 lim x-+0 4x2...
5 answers
1) O3 (CHa)2S2) NaOH HaO+ heatb)
1) O3 (CHa)2S 2) NaOH HaO+ heat b)...
5 answers
Show that if $f$ and $g$ are uniformly continuous on a subset $A$ of $mathbb{R}$, then $f+g$ is uniformly continuous on $A$.
Show that if $f$ and $g$ are uniformly continuous on a subset $A$ of $mathbb{R}$, then $f+g$ is uniformly continuous on $A$....
5 answers
Forl Ithel equation 2x +2y #9xy =0 find the equation of the normal tokthe tangent line at thel point (2,4)
Forl Ithel equation 2x +2y #9xy =0 find the equation of the normal tokthe tangent line at thel point (2,4)...
5 answers
Use cylindrical coordinates_EvaluateJIs_ x + y2 dV, where E is the region that lies inside the cylinder x2 y2 = 4 and between the planes 2 = 0 and 2 = 10,
Use cylindrical coordinates_ Evaluate JIs_ x + y2 dV, where E is the region that lies inside the cylinder x2 y2 = 4 and between the planes 2 = 0 and 2 = 10,...
5 answers
Laura and Nissa are singing duet. Laura sings low note while Nissa sings high note. What different bout their respective sound waves?The (requency the sound waves differenLO The wavelength of the sound waves different0 The spced ol Lhe sound wavcs different
Laura and Nissa are singing duet. Laura sings low note while Nissa sings high note. What different bout their respective sound waves? The (requency the sound waves differenL O The wavelength of the sound waves different 0 The spced ol Lhe sound wavcs different...
5 answers
For the demand equation D(P)= √75-3p, find the elasticity at price p=10. To raise revenue, would you increase or decrease the price?
For the demand equation D(P)= √75-3p, find the elasticity at price p=10. To raise revenue, would you increase or decrease the price?...
5 answers
Griveh A= (finl bas i$ fox Tow Spuce c/ Wmn FO
Griveh A= ( finl bas i$ fox Tow Spuce c/ Wmn FO...
5 answers
Set up but do NOT integrate the Integral which represents the length of the entkze curve
Set up but do NOT integrate the Integral which represents the length of the entkze curve...
5 answers
Aparticle of mass m = ] follows the trajectory given by the time-dependent position vectorrlt) = [cos(St) sin(t)]i + [sin(5t) cos(4t)lj + e-" cos(3t)kCalculate the angular momentum,h,and the moment of force, T about the origin att=0.Angular momentum Calculatc the angular momentum h att = 0: rkserks NcfedQarks ANotedMarks nswrered" Moment of force{ MerksCalculate the moment of force att = 0:
Aparticle of mass m = ] follows the trajectory given by the time-dependent position vector rlt) = [cos(St) sin(t)]i + [sin(5t) cos(4t)lj + e-" cos(3t)k Calculate the angular momentum,h,and the moment of force, T about the origin att=0. Angular momentum Calculatc the angular momentum h att = 0: ...
5 answers
Mount Wrangell Alaska 4317 above sea level (11,2*C). What g*mol - vaph for water 40.79 kJ-mol-1,)boiling point of water the summit? (Hint: Use the barometric formulaPoe~gmh/kBT_ Tne molar massair Is
Mount Wrangell Alaska 4317 above sea level (11,2*C). What g*mol - vaph for water 40.79 kJ-mol-1,) boiling point of water the summit? (Hint: Use the barometric formula Poe~gmh/kBT_ Tne molar mass air Is...
5 answers
Points) Evaluate the integral212 dr = Ir + 2)3
points) Evaluate the integral 212 dr = Ir + 2)3...
5 answers
1 (nd Ir 1 1
1 (nd Ir 1 1...

-- 0.019503--