Which of the following is the inverse relation of the set?<br> {(3, 1), (4, 2), (5, 3), (6,4)}

Let X be a set of size 20 and A CX be of size 10. (a) How many sets B are there that satisfy A Ç B Ç X? (b) How many sets B are

Svetlanka [38]

Answer:

(a) Number of sets B given that

A⊆B⊆C: 2¹⁰. (That is: A is a subset of B, B is a subset of C. B might be equal to C)
A⊂B⊂C: 2¹⁰ - 2. (That is: A is a proper subset of B, B is a proper subset of C. B≠C)

(b) Number of sets B given that set A and set B are disjoint, and that set B is a subset of set X: 2²⁰ - 2¹⁰.

Step-by-step explanation:

Let $x_1, x_2, \cdots, x_{20}$ denote the 20 elements of set X.

Let $x_1, x_2, \cdots, x_{10}$ denote elements of set X that are also part of set A.

For set A to be a subset of set B, each element in set A must also be present in set B. In other words, set B should also contain $x_1, x_2, \cdots, x_{10}$ .

For set B to be a subset of set C, all elements of set B also need to be in set C. In other words, all the elements of set B should come from $x_1, x_2, \cdots, x_{20}$ .

$\begin{array}{c|cccccccc}\text{Members of X} & x_1 & x_2 & \cdots & x_{10} & x_{11} & \cdots & x_{20}\\[0.5em]\displaystyle\text{Member of}\atop\displaystyle\text{Set A?} & \text{Yes}&\text{Yes}&\cdots &\text{Yes}& \text{No} & \cdots & \text{No}\\[0.5em]\displaystyle\text{Member of}\atop\displaystyle\text{Set B?}& \text{Yes}&\text{Yes}&\cdots &\text{Yes}& \text{Maybe} & \cdots & \text{Maybe}\end{array}$ .

For each element that might be in set B, there are two possibilities: either the element is in set B or it is not in set B. There are ten such elements. There are thus $2^{10} = 1024$ possibilities for set B.

In case the question connected set A and B, and set B and C using the symbol ⊂ (proper subset of) instead of ⊆, A ≠ B and B ≠ C. Two possibilities will need to be eliminated: B contains all ten "maybe" elements or B contains none of the ten "maybe" elements. That leaves $2^{10} -2 = 1024 - 2 = 1022$ possibilities.

Set A and set B are disjoint if none of the elements in set A are also in set B, and none of the elements in set B are in set A.

Start by considering the case when set A and set B are indeed disjoint.

$\begin{array}{c|cccccccc}\text{Members of X} & x_1 & x_2 & \cdots & x_{10} & x_{11} & \cdots & x_{20}\\[0.5em]\displaystyle\text{Member of}\atop\displaystyle\text{Set A?} & \text{Yes}&\text{Yes}&\cdots &\text{Yes}& \text{No} & \cdots & \text{No}\\[0.5em]\displaystyle\text{Member of}\atop\displaystyle\text{Set B?}& \text{No}&\text{No}&\cdots &\text{No}& \text{Maybe} & \cdots & \text{Maybe}\end{array}$ .

Set B might be an empty set. Once again, for each element that might be in set B, there are two possibilities: either the element is in set B or it is not in set B. There are ten such elements. There are thus $2^{10} = 1024$ possibilities for a set B that is disjoint with set A.

There are 20 elements in X so that's $2^{20} = 1048576$ possibilities for B ⊆ X if there's no restriction on B. However, since B cannot be disjoint with set A, there's only $2^{20} - 2^{10}$ possibilities left.

5 0

3 years ago

Answer 37, 38, and 39 and make sure to answer with either the letter or with one of the answers shown! SHOW YOUR WORK PLEASE!!

ollegr [7]

Answer:

I didn't even realize until I had written all of the answers down that they are all <u><em>C.</em></u>

37 = C. <u>15%</u>

38 = C. <u>2 $x^{3}$ </u>

39 = C. <u>48</u>

Step-by-step explanation:

37: I used process of elimination to figure out which percentage of decrease it was by just multiplying 42 by each percentage until I got 6.3 which is what you need to subtract from 42 to get 35.7, so the answer is C, or 15%.

38: A coefficient is the number that is being multiplied by the variable, which in this case is "x". So whatever answer involves "2x" is the correct answer. Therefore the answer is C, or <u>2 $x^{3}$ .</u>

39: You have to add the amount of money Henry paid for painting supplies and how much profit he mad to figure out how much money he really made. 400 + 560 = 960 and he charges $20 for each painting so you need to divide 960/20 to get your answer, which is C, or 48.

8 0

3 years ago

Kobe wants to organize his toys. His box has a width 2 2/3 feet, length 3 1/3 feet, and height 2 1/3 feet?

Tanya [424]

Answer:

Cubic feet = Length (3 1/3 feet) x Width (2 2/3 feet) x Height (2 1/3 feet)

Step-by-step explanation:

3 1/3 x 2 2/3 x 2 1/3 = approximately 20.74 cubic feet

3 0

3 years ago

One alloy is 2 parts iron to 3 parts silver and another alloy is 7 parts iron to 3 parts silver. How much of each should be comb

Shalnov [3]

Let x be your first alloy
2/5x+7/10(30-x)=1/2(30)
4x+210-7x=150
3x=60
x=20
So, you need 20 lbs of the first alloy, and 10 parts of the second to make 30 lbs of half iron and half silver alloy!

8 0

4 years ago

A stadium has 48,000 seats. Seats sell for $25 in Section A, $20 in Section B, and $15 in Section C. The number of seats in

Ira Lisetskai [31]

I had a super-ugly equation that didn’t work, but it got me close. (24,000/12,000/12,000 or so)
Then I adjusted the numbers to make the money work.

I’m sorry I can’t help more.

24,000(25) + 14,200(20) + 9,800(15)
= 600,000 + 284,000 + 147,000 = 1,031,000

Section A = 24,000 seats
Section B = 14,200 seats
Section C = 9,800 seats

14,200 + 9,800 = 24,000 seats

5 0

3 years ago

Which of the following is the inverse relation of the set? {(3, 1), (4, 2), (5, 3), (6,4)}