MATHEMATICS COLLEGE

Let A and B be non-empty, bounded subsets of R. (a) Why does sup(AUB) exist? (b) Prove that sup(AUB) = max{sup A, sup B}.

Answers

Answer 1

Answer:

Answer with Step-by-step explanation:

Let A and B be non- empty bounded subset of R

a.We have to find why $sup(A\cup B)$ exist

If A and B are bounded set

Then there exist constant such that

$a\leq A\leq b$ and $c\leq B\leq d$

Then , sup of A =b and sup of B=d

When a set is bounded then all elements lie in the set are lie between the constants s and t.

All elements are less than or equal to t then t is supremum of set.

Because both set are bounded and sup of both set A and B are exist.All elements A union B are less than or equal to sup A or sup B.

$sup(A\cup B)=max(sup A, sup B)$

Then, $sup (A\cup B)$ exist.

b.We have to prove that

$sup (A\cup B)=max(sup A,sup B)$

Suppose ,A =(1,2) and B=(2,3)

Sup A=2 , sup B=3

$(A\cup B)=(1,2)\cup (2,3)$

Upper bound of $A\cup B)=3$

Hence, $Sup (A\cup B)=3$

If A=(4,5),B=(2,3)

Sup A=5,Sup B=3

$A\cup B=(4,5)\cup (2,3)$

$Sup(A\cup B)=5$

Hence, $Sup(A\cup B)=5$

Hence, we can say that $sup(A\cup B)=max(sup A,sup B)$ .

Answer 2

Answer:

Answer:

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Step-by-step explanation:

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(x − 2) is a factor of x4 + 2x3 − 7x2 − 8x + 12. The other factors are

Answers

Answer:

(x - 1) (x - 2) (x + 2) (x + 3)

Step-by-step explanation:

Factor the following:

x^4 + 2 x^3 - 7 x^2 - 8 x + 12

The possible rational roots of x^4 + 2 x^3 - 7 x^2 - 8 x + 12 are x = ± 1, x = ± 2, x = ± 3, x = ± 4, x = ± 6, x = ± 12. Of these, x = 1, x = 2, x = -2 and x = -3 are roots. This gives x - 1, x - 2, x + 2 and x + 3 as all factors:

Answer: (x - 1) (x - 2) (x + 2) (x + 3)

Freida drove 18 miles in 24 minutes. At this rate, how many miles did she drive in 6 minutes?

Answers

Answer:

4.5

Step-by-step explanation: you divide 24 by what you do to get 6 which is 4 then you use 4 to divide 18 which is 4.5 or 4 and a half. Have A Great Day!

A manufacturer knows that their items have a normally distributed length, with a mean of 13.1 inches, and standard deviation of 4.1 inches. If 25 items are chosen at random, what is the probability that their mean length is less than 11.1 inches

Answers

Answer:

0.73% probability that their mean length is less than 11.1 inches

Step-by-step explanation:

To solve this question, we have to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = (X - \mu)/(\sigma)$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean $\mu$ and standard deviation $\sigma$ , a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation, which is also called standard error $s = (\sigma)/(√(n))$