Answer:
(a) and (b) are not true in general. Refer to the explanations below for counterexamples.
It can be shown that (c) is indeed true.
Step-by-step explanation:
This explanation will use a lot of empty sets [tex]\phi[/tex] just to keep the counterexamples simple.
(a)Note that [tex]A \cap B[/tex] can well be smaller than [tex]A[/tex]. It should be alarming that the question is claiming [tex]A\![/tex] to be a subset of something that can be smaller than [tex]\! A[/tex]. Here's a counterexample that dramatize this observation:
Consider:
[tex]A = \left\lbrace 1 \right\rbrace[/tex].[tex]B = \phi[/tex] (an empty set, same as [tex]\left\lbrace \right\rbrace[/tex].)[tex]C = \phi[/tex] (another empty set.)The intersection of an empty set with another set should still be an empty set:
[tex]A \cap B = \left\lbrace 1\right\rbrace \cap \left\lbrace\right\rbrace = \left\lbrace\right\rbrace[/tex].
The union of two empty sets should also be an empty set:
[tex]((A \cap B) \cup C) = \left\lbrace\right\rbrace \cup \left\lbrace\right\rbrace = \left\lbrace\right\rbrace[/tex].
Apparently, the one-element set [tex]A = \left\lbrace 1 \right\rbrace[/tex] isn't a subset of an empty set. [tex]A \not \subseteq ((A\cap B) \cup C)[/tex]. Contradiction.
(b)Consider the same counterexample
[tex]A = \left\lbrace 1 \right\rbrace[/tex].[tex]B = \phi[/tex] (an empty set, same as [tex]\left\lbrace \right\rbrace[/tex].)[tex]C = \left\lbrace 2 \right\rbrace[/tex] (another empty set.)Left-hand side:
[tex](A \cup B) \cap C = \left(\left\lbrace 1 \right\rbrace \cup \left\lbrace \right\rbrace\right) \cap \left\lbrace 2 \right\rbrace\right = \left\lbrace 1 \right\rbrace \cap \left\lbrace 2 \right\rbrace = \left\lbrace \right\rbrace[/tex].
Right-hand side:
[tex](A \cap B) \cup C = \left(\left\lbrace 1 \right\rbrace \cap \left\lbrace \right\rbrace\right) \cup \left\lbrace 2 \right\rbrace\right = \left\lbrace \right\rbrace \cup \left\lbrace 2 \right\rbrace = \left\lbrace 2 \right\rbrace[/tex].
Apparently, the empty set on the left-hand side [tex]\left\lbrace \right\rbrace[/tex] is not the same as the [tex]\left\lbrace 2 \right\rbrace[/tex] on the right-hand side. Contradiction.
(c)Part one: show that left-hand side is a subset of the right-hand side.
Let [tex]x[/tex] be a member of the set on the left-hand side.
[tex]x \in (A \backslash B) \cap C[/tex].
[tex]\implies x\in A \backslash B[/tex] and [tex]x \in C[/tex] (the right arrow here reads "implies".)
[tex]\implies x \in A[/tex] and [tex]x \not\in B[/tex] and [tex]x \in C[/tex].
[tex]\implies (x \in A\cap C)[/tex] and [tex]x \not\in B \cap C[/tex].
[tex]\implies x \in (A \cap C) \backslash (B \cap C)[/tex].
Note that [tex]x \in (A \backslash B) \cap C[/tex] (set on the left-hand side) implies that [tex]x \in (A \cap C) \backslash (B \cap C)[/tex] (set on the right-hand side.)
Therefore:
[tex](A \backslash B) \cap C \subseteq (A \cap C) \backslash (B \cap C)[/tex].
Part two: show that the right-hand side is a subset of the left-hand side. This part is slightly more involved than the first part.
Let [tex]x[/tex] be a member of the set on the right-hand side.
[tex]x \in (A \cap C) \backslash (B \cap C)[/tex].
[tex]\implies x \in A \cap C[/tex] and [tex]x \not\in B \cap C[/tex].
Note that [tex]x \not\in B \cap C[/tex] is equivalent to:
[tex]x \not \in B[/tex], OR[tex]x \not\in C[/tex], ORboth [tex]x \not\in B[/tex] AND [tex]x \not \in C[/tex].However, [tex]x \in A \cap C[/tex] implies that [tex]x \in A[/tex] AND [tex]x \in C[/tex].
The fact that [tex]x \in C[/tex] means that the only possibility that [tex]x \not\in B \cap C[/tex] is [tex]x \not \in B[/tex].
To reiterate: if [tex]x \not \in C[/tex], then the assumption that [tex]x \in A \cap C[/tex] would not be true any more. Therefore, the only possibility is that [tex]x \not \in B[/tex].
Therefore, [tex]x \in (A \backslash B)\cap C[/tex].
In other words, [tex]x \in (A \cap C) \backslash (B \cap C) \implies x \in (A \backslash B)\cap C[/tex].
[tex](A \cap C) \backslash (B \cap C) \subseteq (A \backslash B)\cap C[/tex].
Combine these two parts to obtain: [tex](A \backslash B) \cap C = (A \cap C) \backslash (B \cap C)[/tex].
a soccer organization is going to spend 2990 dollars to purchase all sweatshirts or all jackets for its players. Sweatshirts are 26 dollars each and jackets are 46 dollars each. What is the greatest number of sweatshirts or jackets that the organization can purchase.
Answer:
121,436 to the nearest ten thousand
Step-by-step explanation:
80 plus what equals 44??
Answer:
-36
Step-by-step explanation:
80 + (-36) = 44
The negative in the 36 allows it to be subtracted from 80
Answer:
80 + -36
Step-by-step explanation:
80 - 44 = 36
80 + -36 = 44
Suppose John has 6.93 pounds of seed. If it takes 6.3 pounds of seed to plant one acre
can be planted?
Answer:
he can onle plant one.
Step-by-step explanation:
what exactly is the question.
What is the equation of the line that passes through the point (5,-7) and has a slope of negative 1
Answer:
y = -x - 2
Step-by-step explanation:
Use equation: y = mx + b, where "m" is the slope and "b" is the y intercept. You are given a slope of -1 for the m. This is usually expressed as just a "-", as the 1 is inferred.
Since you are given a point, plug the x and y values in to the equation.
-7 = -(5) + b
Solve for b:
-7 = -5 + b
-2 = b
Knowing "m" and "b", plug these in to get your final equation
y = -x - 2
A square has a side length of 19 in. What is the area?
Answer:
A=a2=192=361in²
Step-by-step explanation:
Stella has 215 coins and 89 of them are nickels. Two-thirds of the remaining coins are pennies. How many pennies does Stella have?
A. 38
B. 42
C. 66
D. 84
Answer: 84
Step-by-step explanation:
215 - 89 = 126
126 divided by 3 = 42
42 x 2 = 84
( this was how my teacher taught me to do these kind of problems)
Hope this helps! <3
Answer:
The answer is 84
Step-by-step explanation:
got it right on edge :D
Why is "3 meals a day" a unit rate?
O It compares 1 meal to 1 day.
O It compares 1 meal to 3 days.
O It compares 3 meals to 1 day.
Olt compares 3 meals to 3 days.
Answer:
it compares 3 meals to 1 day
2. During typhoon Ambo, PAGASA tracks the amount of accumulating rainfall. For the first three hours of typhoon, the rain fell at a constant rate of 25mm per hour The typhoon slows down for an hour and started again at a constant rate of 20 mm per hour for the next two hours. Write a piecewise function that models the amount of rainfall as function of time.
Answer:
Kindly check explanation
Step-by-step explanation:
Given the following information:
First three hours of typhoon: rate of Rainfall. = 25mm per hour
Typhoon slows down for an hour : then Rainfall began again at a constant rate of 20 mm per hour for the next 2 hours
Piecewise function which models Rainfall ad a function of time
Amount of Rainfall as a function of time = f(t)
f(t) = 25t ; if t≤ 3
When typhoon slowed down for an hour :
f(t) = 0;
f(t) = 25*3 ; for t ∈ (3, 4)
Then Rainfall at a constant rate of 20mm for another 2 hours
f(t) = (25 *3) + 20(t - 4) ; for t ∈ (5, 6)
Combine Like Terms: x^2 + y^2 + 3y^2 + 4x + 2x + 2 + 2
Answer:
x^2 + 4y^2 + 6x + 4
Step-by-step explanation:
Arranging the terms;
x ^ 2 + y^2 + 3y^2 + 4x + 2x + 2 + 2
x^2 + 4y^2 + 4x + 2x + 2 + 2
x^2 + 4y^2 + 6x + 2 + 2
x^2 + 4y^2 + 6x + 4
Holiday Inn charges its cost one dollar plus $.80 per minute for long distance calls. Across the street, La Quinta charges its guests $2 plus $0.75 per minute for long distance calls. Which equation correctly models the length of a call for which the two hotels charge the same amount
All side lengths in quadrilateral PQRS measure 6 units. A transformation maps PQRS to P'Q'R'S'. All sides of P'Q'R'S' measure 1 unit. Is the transformation an isometric transformation? Explain
Answer:
no. in an isometric transformation, the side lenths all stay the same. in this transformation (which is a dilation), the size lengths decreased, so it is not isometric.
Step-by-step explanation:
Train A is traveling at 70 mph to a destination 200 miles away. Train B will leave 20 minutes after train A. Train B will leave from Train A's destination and will head in Train A's direction. If Train B will be going at a speed 32 of Train A's, how long after Train A left will it take before Train B arrives at Train A's starting point? Round your answer to the nearest tenth of an hour
Answer:
The answer is "2.2 h"
Step-by-step explanation:
Speed of Train B:
[tex]= \frac{3}{2} \times 70 \ mph \\\\= 105 \ mph[/tex]
Train B is then moving on:
[tex]= \frac{200 \ miles}{ 105 \ mph} \\\\= 1.905 \ h\\[/tex]
Then it takes Train A to the left:
[tex]\to 1.905\ h + 20 \ min \\\\= 1.905 \ h +0.333 \ h \\\\= 2.238 \ h[/tex]
To get train B to the reference point of train A = 2.2 h
A study revealed a strong, linear, positive association between the amount of shampoo used and the length of hair. Another study revealed a strong, linear, positive association between amount of conditioner used and the length of hair. When the data were combined, the association became negative. What is this an example of?
Answer:
Step-by-step explanation:
B
Answer:
Simpson's Paradox
Step-by-step explanation:
Which statement is true?
O A. The absolute value of 3 is -3.
OB. The opposite of 7 is 7.
O C. The absolute value of 5 is equal to the opposite of 5.
O D. The opposite of -4 is
equal to the absolute value of -4.
Answer:
D. The opposite of -4 is equal to the absolute value of -4.
Step-by-step explanation:
If m 24 = 35º, find m_2 and m23.
А
00
2
4
.
m22 =
m23 =
Answer:
m22 answer will be in your book only
Jefferson Middle School needs to see a 300% gain in PTO fundraiser
sales from last year. If they sold a total of $2,420 last year, how much do
they need to sell this year to meet their goal? *
Given parameters:
Percentage gain = 300%
Selling cost last year = $2420
Unknown:
Selling cost this year = ?
Solution:
To solve this problem;
Profit percent = [tex]\frac{Selling cost this year - Selling cost last year}{Selling cost last year} x 100[/tex]
300 = [tex]\frac{S - 2420}{2420} x 100[/tex]
3 = [tex]\frac{S - 2420}{2420}[/tex]
S- 2420 = 7260
S = $9680
To make a 300% gain, Jefferson Middle school needs a $9680 from fundraiser sales this year.
The "cold start ignition time" of an automobile engine is being investigated by a gasoline manufacturer. The following times (in seconds) were obtained for a test vehicle: 1.75, 1.92, 2.62, 2.35, 3.09, 3.15, 2.53, and 1.91. (a) Calculate the sample mean and sample standard deviation. (b) Construct a box plot of the data.
Answer:
(a) The sample mean and sample standard deviation are 2.415 and 0.5342 respectively.
(b) Shown below.
Step-by-step explanation:
The data set provided is:
S = {1.75, 1.92, 2.62, 2.35, 3.09, 3.15, 2.53, and 1.91}
(a)
Compute the sample mean and sample standard deviation as follows:
[tex]\bar x=\frac{1}{n}\sum\limits_{i}{x_{i}} =\frac{1}{8}\times 19.32=2.415\\\\\\s=\sqrt{\frac{1}{n-1}\sum\limits_{i}{(x_{i}-\bar x^{2})^{2}}}=\sqrt{\frac{1}{8-1}\times 1.9976}=0.5342[/tex]
Thus, the sample mean and sample standard deviation are 2.415 and 0.5342 respectively.
(b)
Use SPSS to form a Box plot.
Go to Graphs → Legacy Dialogs → Boxplot
A dialog box will open.
Select "Simple".
Select "Summaries for Separate Variable"
Press "Define"
Another dialog box will open.
Select the variable.
Press OK.
The box plot is attached below.
Solve the math problem. Make sure your answer has the correct number of significant figures.
10.9 - 8.264
Answer:
2.636
Step-by-step explanation:
10.900-08.264
Sophie went to the lake with her family. First, they spent 1 1/4
hours fishing. Then, they spent twice as much time swimming. How much time did they spend fishing and swimming in all?
Write your answer as a whole number, fraction, or mixed number.
Answer:
2
Step-by-step explanation:
Find the axis of symmetry for the function below.
2x^2 – 8x + 5 = 0
Which graph shows the solution to this system of inequalities? 3x - y < 5 and x + y > 2( PLEASE HELP!! )
Answer:
D
Step-by-step explanation:
Just graph both inequalities and plug in 0 to see which way it goes.
The graph of a linear function is shown on the grid. What is the rate of change of y with respect to x for this function? Please hurry it’s urgent Thank you
Answer:
[tex] -0.2 [/tex]
Step-by-step explanation:
Given:
(-3, 3.6);
(5, 2)
Required:
Rate of change
SOLUTION:
Rate of change for the function is given as [tex] \frac{y_2 - y_1}{x_2 - x_1} [/tex].
Let,
[tex] (-3, 3.6) = (x_1, y_1) [/tex]
[tex] (5, 2) = (x_2, y_2) [/tex]
Rate of change = [tex] \frac{2 - 3.6}{5 - (-3)} [/tex]
[tex] = \frac{-1.6}{8} [/tex]
[tex] = -0.2 [/tex]
The rate of change of y with respect to x for this function is -0.2
Given :
The graph of a linear function is shown on the grid
To find the rate of change of y with respect to x , we need to pick two points from the graph
Two points are (-3,3.6) and (5,2)
Lets apply slope formula to find out rate of change
[tex]slope =\frac{y_2-y_1}{x_2-x_1}\\slope =\frac{2-3.6}{5+3} \\slope = -0.2[/tex]
The rate of change of y with respect to x for this function is -0.2
Learn more : brainly.com/question/24930128
Latoya has a bookcaes with k shelves. There are 14 books on each shelf. Using k, write an expression for the total number of books
Answer:
Step-by-step explanation:
Pamela has 15 different books. In how many ways can she place her books on two selves so that there is at least one book on each shelf? (Consider the books in each arrangement to be stacked one next to the other, with the first book on each shelf at the left of the shelf.)
Initially, I thought that this would be fairly straight forward. There are two cases where either shelf could be empty and so we would subtract 2 from the total number of permutations which is obviously 15!. It is obvious now that both of these statements are false.
My previous thoughts didn't take into account the fact that there are two shelves and that the books could be stacked in such a way that 5 are on the top shelf and 10 are on the bottom, or 8 on top with 7 on bottom, or any number of other ways. I believe there are 214 ways the books can be distributed amongst the two shelves (keep in mind that each shelf has to have at least one book, 15−1=14) if the ordering of the books didn't matter, but it does. Is this correct?
Help please.... I have got this wrong 2xs only once chance left
Answer:
a: 5x-1 = -6
b: 5x-1 = -6
+1 +1
5x = -5
divide
x = -1
Complete the following truth table. Use T for true and F for false.
what is the answer
Convert 1_1/2 to a percent
Answer:
150%
Step-by-step explanation:
You have 1 hole 100% and half 50%
Answer:the fraction 48 can be converted to decimal by dividing 4 by 8 . It can be converted to percent by multiplying by 100
Step-by-step explanation:
all the manz in my block have a glock ahhhhhh
1/2 (b + 2) + 3b = -1
Answer:
3 1/2b + 2 1/2=-1
Step-by-step explanation:
someone help please
Answer:
16. =
17. <
18. <
Step-by-step explanation:
Dilations always increase the length of line segments
True
False
Dilations increase the measure of angles.
Group of answer choices
True
False
Dilations of a triangle are similar to the original triangle.
Group of answer choices
True
False
Write an addition problem that has a sum of -4 3/5
Answer:
-4+(-4.6)
Step-by-step explanation:
