Give examples, if possible, of the following.
i) A set with a supremum but no maximum.
ii) A decreasing sequence so that does not exist
iii) An increasing sequence so that does not exist.
Give examples, if possible, of the following. i) A set with a supremum but no maximum....
Separate each answer? 5) Define the supremum of a bounded above set SCR. 6) Define the infimum of a bounded below set SCR. 7) Give the completeness property of R 8) Give the Archimedean property of R. 9) Define a density set of R. 10) Define the convergence of a sequence of R and its limit. 11) State the Squeeze theorem for the convergent sequence. 12) Give the definition of increasing sequence, decreasing sequence, monotone se- quence. 13) Give the...
Let be an arbitrary function and A X. i) Show that A ii) Give an example to show that in general A = . iii) Show that, if is injective, then A = iv) Show that, if X and Y are modules; is a homomorphism of modules and A is a submodule of X such that ker, then we also have A = We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe...
Let be i.i.d. . Define the sample mean and the sample variance by and . (i) Find the distribution of and for i = 1, ... , n. (ii) Show that and are independent for i = 1, ... , n. (iii) Hence, or otherwise, show that and are independent. 7l N (μ, σ2) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let be a prime and let be the set of rational numbers whose denominator (when written in lowest terms) is not divisible by . i) Show, with the usual operations of addition and multiplication, that is a subring of . ii) Show that is a subring of . iii) Is a field? Explain. iv) What is where is the set of all fractions with denominator a power of We were unable to transcribe this imageWe were unable to transcribe this...
Give three examples for Rolle's Theorem: For the first, define f : [0, 1] R such that condition 1 does not hold, condition 2 does hold, condition 3 does hold, and f'(c)0 for every c (0,1). For the second example, make sure only condition 2 does not hold and the conclusion do not hold. For the third example, do the same with condition 3. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
Show that if is the cycle . Give examples. What does this say for 2-cycles? We were unable to transcribe this image(a, a, ..., and then 8-1 = (an, an-1, ..., 1)
3. Let ,..., be independent random sample from N(), where is unknown. (i) Find a sufficient statistic of . (ii) Find the MLE of . (iii) Find a pivotal quantity and use it to construct a 100(1–)% confidence interval for . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
The figure below shows a graph of the derivative of a function . Use this graph to answer parts (a) and (b) (a) On what intervals is increasing or decreasing? (b) For what values of does have a local maximum or minimum? (It asks to be specific). Only the values are needed (not ordered pairs). We were unable to transcribe this imageWe were unable to transcribe this imagepe & Bl apr derivative f' of a function f. Use this graph...
If are commutative rings, define their direct product by induction on ( it is the set of n- tuples ( ) with for all i). Prove that the ring where is the set with is the direct product of copies of . R1, ..., Rn R1 X ... X Rn n> 2 We were unable to transcribe this imageTi ER We were unable to transcribe this imageWe were unable to transcribe this imageX = n, We were unable to transcribe...
Let a and be be in . Show the following. If gcd(a,b)=1, then for every n in there exist x and y in such that n=ax+by. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image