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wlad13 [49]
3 years ago
6

Decide which of the following represent true statements about the nature of set. For any that are false, provide a specific exam

ple where the statement in question does not hold.
(a) If A1 ⊇ A2 ⊇ A3 ⊇ A4 ... are all sets containing an infinite number of elements, then the intersection n-1 An is infinite as well.
Mathematics
1 answer:
mart [117]3 years ago
8 0

Answer:

If the intersection is finite the statement  is true, but if the intersection is infinite the statement is false.

Step-by-step explanation:

From the statement of the problem I am not sure if the intersection is finite or infinite. Then, I will study both cases.

Let us consider first the finite case: A = \cap_{i=1}^{n}A_i. Because the condition A1 ⊇ A2 ⊇ A3 ⊇ A4 ... we can deduce that the set A_n is a subset of each set A_i with i\leq n. Thus,

\cap_{i=1}^{n}A_i = A_n.

Therefore, as A_n is infinite, the intersection is infinite.

Now, if we consider the infinite intersection, i.e. A = \cap_{k=1}^{\infty}A_k the reasoning is slightly different. Take the sets

A_k = (0,1/k) (this is, the open interval between 0 and 1/k.)

Notice that (0,1) ⊇ (0,1/2) ⊇ (0, 1/3) ⊇(0,1/4) ⊇...So, the hypothesis of the problem are fulfilled. But,

\cap_{k=1}^{\infty}(0,1/k) = \empyset

In order to prove the above statement, choose a real number x between 0 and 1. Notice that, no matter how small x is, there is a natural number K such that 1/K. Then, the number x is not in any interval (0,1/k) with k>K. Therefore, x is not in the set [tex]\cap_{k=1}^{\infty}(0,1/k)[\tex].

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brilliants [131]

Answer:

80

Step-by-step explanation:

There are four 4 by 3 rectangles, so I did 4x3 and got 12, then multiplied by 4 to get 48.  There are also two 4 by 4 squares, so 4x4=16 and 16x2=32

32+48=80

8 0
3 years ago
Students are making ornaments and each takes 1 2 of a piece of Bristol board. They go to the cupboard and find 1 1 2 pieces of B
s2008m [1.1K]

Answer: 3

Step-by-step explanation:

Feom the question, we are informed that students are making ornaments and each takes 1/2 of a piece of Bristol board and that they go to the cupboard and find 1 1/2 pieces of Bristol board.

The number of ornaments that they can make will be calculated by dividing 1 1/2 by 1/2. This will be:

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8 0
2 years ago
How do you solve 8x-7= 3x 9?
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3 0
2 years ago
17-(5-p)=2(5p-16)<br> Step by step explanation please
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Answer:

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Step-by-step explanation:

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2 years ago
select the irrational number? A)√9 B)√12 C)√16 D)√20 E)√25 (there should be more than one answer)
GenaCL600 [577]
The answers are:
__________________________
    [B]:  "√12 " ;   AND:
__________________________
    [D]:  "<span>√20" .
__________________________
Note:  "Irrational numbers" have decimals that are:  1) non-terminating;  AND:  2) non-repeating.
__________________________
Consider the answer choices given; and consider the given information that there "should be no more than one answer.
__________________________________
Consider the following answer choices:
__________________________________
Choice A)  "</span><span>√9"  =  3 .  Rule out this choice.
__________________________________
Choice C)  "</span><span>√16" = 4 .  Rule out this choice.
__________________________________
Choice E)  "</span><span>√25" = 5.   Rule out this choice.
_________________________________
We are left with 2 (two) remaining answer choices:
    [B]:  "</span>√12"  ;  and:  [D]:  "<span>√20" .
_________________________________
If there are more than one answer, then these two choices should be the correct answer.   However, let us explore these two choices, using a calculator.
____________________________
    [B]:  </span>"√12"  =  3.4641016151377546

    [D]:  "√20"  =  4.4721359549995794
_________________________________
These two answer choices should the correct answers.
6 0
3 years ago
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