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Valentin [98]
3 years ago
14

A and B are bounded non-empty subsets of R. For inf(A) to be less than or equal to inf(B), which of the following conditions mus

t be met?
a) For every b in B and epsilon > 0, there exists a in A, such that a < b + epsilon.

b) There exists a in A, and b in B such that a < b.

If neither of these conditions are appropriate, what would be appropriate conditions for inf(A) to be less than or equal to inf(B)?
Mathematics
1 answer:
nirvana33 [79]3 years ago
6 0

Answer:

a) must be met

Step-by-step explanation:

We have two conditions:

a) For every b\in B and \epsilon>0, there exists a\in A, such that a.

b) There exists a\in A and b\in B such that  a.

We will prove that conditon a) is equivalent to inf(A)\leq inf(B)

If a) is not satisfied, then it would exist b\in B and \epsilon >0 such that, for every a\in A, a\geq b+\epsilon. This implies that b+\epsilon is a lower bound for A and in consequence

inf(A)\geq b+\epsilon > b\geq inf(B)

Then, inf(A) \leq inf(B) implies a).

If inf(A) \leq inf(B) is not satisfied then, inf(A) > inf(B) and in consequence exists b\inB such that b-inf(A)=\epsilon >0. Then b-\epsilon=inf(A) and, for every a\in A,

b-\epsilon =inf(A)\leq a.

So, a) is not satisfied.

In conclusion, a) is equivalent to inf(A)\leq inf(B)

Finally, observe that condition b) is not an appropiate condition to determine if inf(A)\leq inf(B) or not. For example:

  • <u>A={0}, B={1}</u>. b) is satisfied and inf(A)=0
  • <u>A={0}. B={-1,1}</u>. b) is satisfied and inf(A)=0>-1=inf(B)

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find the area of the triangle. please check my answer!! i need it done now! :( no one will help ive reposted it a few times now
mina [271]

Answer:

A = 437 mm²

Step-by-step explanation:

The general formula for the area of a triangle:

A = \frac{1}{2}bh, where 'b' is the measure of the base and 'h' is the measure of the height

Given the dimensions of the triangle in the picture:

A = \frac{1}{2}(38)(23) = 437 mm²

7 0
3 years ago
Parallel / Perpendicular Practice
deff fn [24]

The slope and intercept form is the form of the straight line equation that includes the value of the slope of the line

  1. Neither
  2. ║
  3. Neither
  4. ⊥
  5. ║
  6. Neither
  7. Neither
  8. Neither

Reason:

The slope and intercept form is the form y = m·x + c

Where;

m = The slope

Two equations are parallel if their slopes are equal

Two equations are perpendicular if the relationship between their slopes, m₁, and m₂ are; m_1 = -\dfrac{1}{m_2}

1. The given equations are in the slope and intercept form

\ y = 3 \cdot x + 1

The slope, m₁ = 3

y = \dfrac{1}{3} \cdot x + 1

The slope, m₂ = \dfrac{1}{3}

Therefore, the equations are <u>neither</u> parallel or perpendicular

  • Neither

2. y = 5·x - 3

10·x - 2·y = 7

The second equation can be rewritten in the slope and intercept form as follows;

y = 5 \cdot x -\dfrac{7}{2}

Therefore, the two equations are <u>parallel</u>

  • ║

3. The given equations are;

-2·x - 4·y = -8

-2·x + 4·y = -8

The given equations in slope and intercept form are;

y = 2 -\dfrac{1}{2}  \cdot x

Slope, m₁ = -\dfrac{1}{2}

y = \dfrac{1}{2}  \cdot x - 2

Slope, m₂ = \dfrac{1}{2}

The slopes

Therefore, m₁ ≠ m₂

m_1 \neq -\dfrac{1}{m_2}

The lines are <u>Neither</u> parallel nor perpendicular

  • <u>Neither</u>

4. The given equations are;

2·y - x = 2

y = \dfrac{1}{2} \cdot   x +1

m₁ = \dfrac{1}{2}

y = -2·x + 4

m₂ = -2

Therefore;

m_1 \neq -\dfrac{1}{m_2}

Therefore, the lines are <u>perpendicular</u>

  • ⊥

5. The given equations are;

4·y = 3·x + 12

-3·x + 4·y = 2

Which gives;

First equation, y = \dfrac{3}{4} \cdot x + 3

Second equation, y = \dfrac{3}{4} \cdot x + \dfrac{1}{2}

Therefore, m₁ = m₂, the lines are <u>parallel</u>

  • ║

6. The given equations are;

8·x - 4·y = 16

Which gives; y = 2·x - 4

5·y - 10 = 3, therefore, y = \dfrac{13}{5}

Therefore, the two equations are <u>neither</u> parallel nor perpendicular

  • <u>Neither</u>

7. The equations are;

2·x + 6·y = -3

Which gives y = -\dfrac{1}{3} \cdot x - \dfrac{1}{2}

12·y = 4·x + 20

Which gives

y = \dfrac{1}{3} \cdot x + \dfrac{5}{3}

m₁ ≠ m₂

m_1 \neq -\dfrac{1}{m_2}

  • <u>Neither</u>

8. 2·x - 5·y = -3

Which gives; y = \dfrac{2}{5} \cdot x +\dfrac{3}{5}

5·x + 27 = 6

x = -\dfrac{21}{5}

  • Therefore, the slopes are not equal, or perpendicular, the correct option is <u>Neither</u>

Learn more here:

brainly.com/question/16732089

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Mr. Wilson is giving his students a vocabulary test tomorrow over all the terms they have learned from this semester. Today, he
Y_Kistochka [10]
The population in the above situation is the total number of students under Mr. Wilson.

A sample is a part of the population that may best represent the population. There is no sample  in the above situation because Mr. Wilson made all students pick 5 note cards. He will be able to determine the performance of each student based on their performance in picking 5 note cards and defining the terms in each card.




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Which of the following sets of numbers could represent the three sides of a right<br> triangle?
sweet-ann [11.9K]

Answer:

(45,60,75)

Step-by-step explanation:

Because 45^2 + 60^2 = 75^2

                  5625 = 5625

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2 years ago
Raquel works in a pet store every day after school and sometimes on weekends. She earns $5.20 per hour. If Raquel works 21 hours
riadik2000 [5.3K]

Answer:

B

Step-by-step explanation:

$5.20 is about $5.00, 21 is about 20, so $5.00 * 20 *4 = $400.

Checking this: $5.20 * 21 * 4 = $436.80

8 0
2 years ago
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