Which of the following statements is true for all sets A,B and C? Give a proof
or a counter example.
(a) A ⊆ ((A∩B)∪C).
(b) (A∪B)∩C = (A∩B)∪C.
(c) (A\B)∩C = (A∩C) \ (B∩C).
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].
John is saving money for a digital camera
that cost $490. So far, he saved $175. If
he will save $50 per week, how many more
weeks does he need to save to buy the
camera? Which of the following
represents the right inequality?
Answer
7 weeks
Step-by-step explanation:
$490 - $175 = $315
John needs $315 still for the camera
divdie $315 by $50 and you get 6.3 but because it cant be 6 weeks he would be short $15 you would have to wait an additional week in order to cover the cost of the camera completly.
Brainliest please!!!!
What is the slope of the line that contains these points?
(5,-3),(6,-1),(7,1),(8,3)
Answer:
2
Step-by-step explanation:
slope of line= [tex]\frac{y2-y1}{x2-x1}[/tex]
You can take any two points from the given points, put them in this formula and answer will be 2.
For example:
Lets take (7,1) and (8,3)
Slope= [tex]\frac{3-1}{8-7}[/tex]
=2/1=2
Hope it will help :)
Y’all pls help me i have to turn this tonight
A study was conducted to analyze the amount of time spent sitting at a desk for employees over a work day. The time spent sitting was recorded for a large sample of employees and found to follow a bell-shaped distribution with a mean of about 280 minutes. The smallest amount of time was 100 minutes and the largest amount of time was 460 minutes. Unfortunately, the standard deviation was not reported in the summary. Which of the following represent the most appropriate value for that standard deviation?
A. Options 60 minutes
B. 90 minutes
C. 180 minutes
D. 360 minutes
Find the value of x
[tex]8x + 40 = 120[/tex]
Subtract sides -40
[tex]8x + 40 - 40 = 120 - 40[/tex]
[tex]8x = 80[/tex]
Divided sides by 8
[tex] \frac{8}{8}x = \frac{80}{8} \\ [/tex]
[tex]x = 10[/tex]
Done .....♥️♥️♥️♥️♥️
Based on the information marked in the diagram, ABC and DEF must be congruent.
True Or False?
Applying the HL congruence theorem, the information marked in the diagram cannot be congruent, so it is FALSE.
What is the HL Congruence Theorem?The HL congruence theorem states that if the corresponding legs of two right traingles are congruent, and the their hypotenuse are also congruent, then the triangles are congruent.
From the information given, we are not told if the hypotenuse of the right triangles are congruent, so we can't apply the HL congruence theorem to prove they are congruent. The answer is FALSE.
Learn more about the HL congruence theorem on:
https://brainly.com/question/2102943
Qs bisect PQR and MRSQ =71
Answer: If ∡PQR = 82°, and the ray QS bisects it, it cuts ∡PQR into two equal halves, ∡PQS and ∡RQS, each of which is then 82/2, or 41°.
Determine the intercepts of the line
X-intercept: (__ , __)
Y-intercept: (__, __)
Solve the following inequality -9/2x grater then or equal to -1/4
Answer:
Did you mean [tex]\frac{-2x}{9} ?\\[/tex]
Please clarify so that I can work this problem out for you :)
Step-by-step explanation:
whats 9×9 how do you get the answer
Answer:
9 × 9 = 81
This is basic 12x12 tables math
Answer:
81
Step-by-step explanation:
you just take 9 and times it by 8 and you get 81
Plz help I have this and i am being timed
Answer:
i think it is -1/2
Answer:
[tex] -\dfrac{13}{18} [/tex]
Step-by-step explanation:
[tex] -\dfrac{8}{9} - (-\dfrac{1}{6}) = [/tex]
[tex] = -\dfrac{8}{9} + \dfrac{1}{6} [/tex]
[tex] = -\dfrac{16}{18} + \dfrac{3}{18} [/tex]
[tex] = -\dfrac{13}{18} [/tex]
To find 2.43 x 7.5, multiply__*_. Then, place the decimal point in the product to show
decimal places. PLEASE BRAINILY HELPPP
Answer:
16.825 is the answer for your question
Use the law of detachment to determine what you can conclude from the given information if three points lie on the same line then they are collinear points A B and C lie on the line M
Answer: Points A,B, and C are collinear
Step-by-step explanation:
Alyssa wants to put a wallpaper border around the top of her bedroom walls. Her room is rectangular and is 12ft long, 11 ft wide, and 8 ft tall.
Answer:
Alyssa needs 1056 cubed feet of wallpaper.
Step-by-step explanation:
Its just 12 times 11 times 8 to find the volume.
what is d divided by equals
Answer: X and Y
Step-by-step explanation: IF I REMEMBER dont get mad if you don’t get it right
Amy practice yoga 20 minutes each day before school and 30 minutes each day after school she is in school Monday through Friday how many minutes does Amy practice yoga in four weeks please show your work
Answer:
1000
Explanation:
The equation would be 5(20+30)*4=_____
First, we add 20 and 30 to get 50, now we multiply 5 by 50 to get 250, then multiply 250 by 4 to get 1000.
Can I also get a Brainliest?
Kayla thinks that the slope of a vertical line is undefined, while Joshua
argues that the slope of a vertical line is zero. Who is correct?
a. Kayla b. Joshua
Answer:
Kayla
Step-by-step explanation:
Vertical lines are always undefined because if your answer has to be divided by zero, the answer is an undefined operation.
Horizontal lines always have the slope of zero!
The slope of a vertical line is undefined.
Kayal is correct.
The rate of change in the table is -6.
The slope of the graph is 3/2.
What is a slope?The slope of a is the change in the y coordinate with respect to the change in the x coordinate.
We have,
A vertical line means there is no change in the x-coordinate.
i.e slope = change in y-coordinates / change in x-coordinate
slope = change in y-coordinate / 0
slope = undefined
The slope of a vertical line is undefined.
Now,
From the given table we see that there is a change of 1 for the x-values and a change of -6 in the y-values so,
Rate of change in the table:
= change in y values/change in the x values
= -6 / 1
= -6
Now,
Take any two points on the graph such as:
(a, b) = (4, 6) and (c, d) = (-4, -6)
The slope of the graph:
= (d - b) / (c - a)
= -6 - 6 / -4 - 4
= -12 / -8
= 3 / 2
Thus,
The slope of a vertical line is undefined.
Kayal is correct.
The rate of change in the table is -6.
The slope of the graph is 3/2.
Learn more about slope here:
https://brainly.com/question/26650191
#SPJ5
Mrs. Smith rode the bus 720 miles in 3 days. On the first day, she traveled 3 times as far as she did on the second day. On the third day, she traveled 2 times as far as she did on the second day. How far did she travel each day?
The first day = 3x
The second day = x
The third day = 2x
720 = 3x + x + 2x
Combine like terms.
720 = 6x
Divide both sides by 6
120 = x
The first day = 3x = 3(120) = 360 miles
The second day = x = 120 miles
The third day = 2x = 2(120) = 240 miles
A 9 foot long section of Kryptonite weighs 1673 pounds. What is the weight of a 1 foot long section of a Kryptonite?
ent
Use the interactive to find an expression equivalent to
4x - 4 - 2x + 3.
O 2x - 1
O 6x - 1
O 2x - 7
O-2x + 3
Please hurry help
Answer:
it's A
Step-by-step explanation:
2x - 1
I put b and got it wrong
Please help thank you very much!!
Answer:
umm...
D...765 4th over 5th
Help? Only right answers get thanks and five stars.
Sloan was told to written paper of a science experiment should Express measurements and leaders 1.26 gallon 11.05 courts 14.2 to cups which shows how to calculate the number of liters he use if he use 5 cups of water divided by 5 4.22 divide 5 by 1.05 or multiply 5 by 1.05
Answer:
Multiply 5 by 4.22 because 5 cups of water is 4.22 liters
Solve and graph 8(p+3)>2(4p-1)
There are 11 frogs at the pond. There are 5 frogs in the water. The rest are in the mud. How many frogs are in the mud?
Answer:
This is easy math. 11-5=6. Six frogs are in the water.
Step-by-step explanation:
11 represents the whole of frogs, and 5 represents the frogs in the water.
Find the equation, function and describe. (100 points)
X
1
2
3
4
5
6
7
Y
5
2.5
1.6667
1.25
1
.83333
.71429
Answer:
257890 is the function of the equation
Step-by-step explanation:
mark me BRAINLIEST
Write an equation in slope-intercept form for the line that passes through (1,-2) and (3, 7)
Answer:
y=9/2x -13/2
Step-by-step explanation:
i used a calculator i hope its right
Please help me! I really need help with this question!! I need 1-19! I don’t need 2 though
Answer:
1) 122°
2) 58°
3) 60°
4) 120°
5) 58°
6) 62°
7) 60°
8) 58°
9) 62°
10) 60°
11) 58°
12) 122°
13) 120°
14) 60°
15) 62°
16) 58°
17) 122°
18) 122°
19) 58°
Step-by-step explanation:
Basically Angle Relationships
Each Angle all matches to a different Angle due to their same degree. It is kind of confusing at first but really easy when you understand it and look at it with pacience.
This took so long lol. I really hope it helps! :D
Are these relations functions?
Answer:
Letter B is yes because if you use the vertical line test, never anytime the vertical line intersect the graph in more than one point.
