prove that (A') '=A by arbitrary elementary method
1 Elementary Set Theory
Notation:
{} enclose a set.
{1, 2, 3} = {3, 2, 2, 1, 3} because a set is not defined by order or multiplicity.
{0, 2, 4, . . .} = {x|x is an even natural number} because two ways of writing
a set are equivalent.
∅ is the empty set.
x ∈ A denotes x is an element of A.
N = {0, 1, 2, . . .} are the natural numbers.
Z = {. . . , −2, −1, 0, 1, 2, . . .} are the integers.
Q = {
m
n
|m, n ∈ Z and n 6= 0} are the rational numbers.
R are the real numbers.
Axiom 1.1. Axiom of Extensionality Let A, B be sets. If (∀x)x ∈ A iff x ∈ B
then A = B.
Definition 1.1 (Subset). Let A, B be sets. Then A is a subset of B, written
A ⊆ B iff (∀x) if x ∈ A then x ∈ B.
Theorem 1.1. If A ⊆ B and B ⊆ A then A = B.
Proof. Let x be arbitrary.
Because A ⊆ B if x ∈ A then x ∈ B
Because B ⊆ A if x ∈ B then x ∈ A
Hence, x ∈ A iff x ∈ B, thus A = B.
Definition 1.2 (Union). Let A, B be sets. The Union A ∪ B of A and B is
defined by x ∈ A ∪ B if x ∈ A or x ∈ B.
Theorem 1.2. A ∪ (B ∪ C) = (A ∪ B) ∪ C
Proof. Let x be arbitrary.
x ∈ A ∪ (B ∪ C) iff x ∈ A or x ∈ B ∪ C
iff x ∈ A or (x ∈ B or x ∈ C)
iff x ∈ A or x ∈ B or x ∈ C
iff (x ∈ A or x ∈ B) or x ∈ C
iff x ∈ A ∪ B or x ∈ C
iff x ∈ (A ∪ B) ∪ C
Definition 1.3 (Intersection). Let A, B be sets. The intersection A ∩ B of A
and B is defined by a ∈ A ∩ B iff x ∈ A and x ∈ B
Theorem 1.3. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Proof. Let x be arbitrary. Then x ∈ A ∩ (B ∪ C) iff x ∈ A and x ∈ B ∩ C
iff x ∈ A and (x ∈ B or x ∈ C)
iff (x ∈ A and x ∈ B) or (x ∈ A and x ∈ C)
iff x ∈ A ∩ B or x ∈ A ∩ C
iff x ∈ (A ∩ B) ∪ (A ∩ C)
Does the last sentence make sense? Or is the grammar not correct?
(In the photo)
Answer:
The grammar is not correct. Replace is with are.
Explanation:
Identities are. Plural subject takes plural verb.
HELP ME PLEASE Due today will give crown please
Discuss two practices of the Eightfold Path. How are they important for our altruism?
Either of the two questions on the Bodhisattva may use additional concepts such as karma.
Answer:
The Eightfold Path consists of eight practices: right view, right resolve, right speech, right conduct, right livelihood, right effort, right mindfulness, and right samadhi ('meditative absorption or union')
Explanation:
The Noble Eightfold Path (also called the Middle Way, or the Threefold Way) is the fourth part (Magga) of the Four Noble Truths . It gives Buddhists a path they can follow to end suffering
How do fishermen take part in the “destruction of life” in the ocean? How do you know?
Answer:
Fishermen take part in the destruction of life by killing fish.
Explanation:
c : the place where you play games like tennis.
Answer:
Badminton. Just like Tennis, Badminton is a game that requires rackets.
Pickleball. Of course our favorite choice Pickleball
Squash. Squash is all about playing indoors or somewhere enclosed.
Beach Tennis.
Racquetball.
Soft Tennis.
Platform Tennis.
Basque Pelota
The romantic movement in literature began as a reaction to the what movement
Answer:
Explanation:
The Romantic movement in literature began as a reaction to the Enlightenment Movement. Every movement in literature is a reaction to its previous era. That is why Romanticism is a reaction to Enlightenment, and Realism is a reaction to Romanticism, and Modernism is a reaction to Realism, and so on.
What does this excerpt reveal about Remeo’s character?
Answer:
it reveals that Romeo is a lover not a fighter he tries his best to stay out of way but isn't always successful. He wants to be a peaceful person but feels that he is forced to defend himself in the situation.
Explanation:
(hope this helps)
_moment when I heard / was accepted to the university, but I know I was very excited.
I don't remember the
a. precise
b.fundamental
c. vibrant
d. paramount
A
Answer:
A.
Explanation:
The correct word to complete the given sentence would be 'precise.'
The term precise is derived from the Latin word praecis which means 'cut short.' The word precise means something that is marked with accuracy and exactness.
The speaker in the given statement is asserting that he/she doesn't remember that exact moment when he/she heard about being accepted to the university, but remembers being excited upon hearing the news.
Therefore, option A is correct.
1.The receptionist_________me when I________for the room keys.
A) didn't understand /ask
B) wasn't understand /asked
C) wasn't understanding /asked
D)understood/ asked
2.Sheila :______you ______
Matt today?
Jim: Yes. He was at the cafe.
A)were / seeing
B) did / see
C) Do /see
D) were / see
3. When______she_____back from London ?
A) did / come
B) was/come
C) did / coming D) was/came
4.Türkan Saylan was a very important Turkish dermatologist and an activist She______the Istanbul Lebrosy Hospital in 1981.For many years,she worked_______and visited Anatolian cities to cure sick people.
A) provided /unfairly B) came up with / unfair C) trained / genius D) founded /voluntarily
5. Living in a/an________society teaches many important things. There are different people and various ideas from many different countries, races or religions. By this way, you can understand people from different
________ better.
A) ancient / generation B) unique /nations C) multicultural /nations D)mystic / generations
6.Driving_________be very dangerous in the past because there_______be seat belts.
A) used to / didn't use to B)didn't use to /use to C) used to use to D) didn't use to didn't use to
Answer:
1) D
2) B
3) A
4) D
5) B
6) A
Explanation:
Base on grammar.
Need help please, it's urgent. about organizational structure of prose. Thank you.
The summary of girl who couldn't see herself by leena dhingra
Answer:
Sounds like an interesting story but I would need to see the passage to answer this haha.
What evidence could be added to the passage to support jill’s infrence
Answer:
Quotes from the text
Explanation:
I’m just trying to earn points I don’t really know what I’m doing
in a group discussion, which of these is the best example of a clarifying question
Answer:
D. When you say H. G. Wells lacked genius, how are you defining "genius"?
If that's one of your choices, because out of the four options, option letter D, presents the example of a question that is asking to clarify upon the meaning of a word, in this case, the word "genius". Clarifying information is important to guarantee the understanding of the idea.
What is the main idea of this passage? the artist's search
Answer:
The search for his art
Explanation:
what is standardization mark of Frozen Peas?
Answer:
Grade C. Inspection Aid 106 Universal Sizer, Inspection Aid 107 USDA Pea Sizer.
Explanation:
I think so...
The lion was roaring loudly. The lion jumped into the air to chase its prey.
Which revision uses a participial phrase to combine these two sentences correctly?
As the lion was roaring loudly, it jumped into the air to chase its prey.
Roaring loudly, the lion jumped into the air to chase its prey.
The lion was roaring loudly and jumped up to chase its prey.
When the lion jumped in the air to chase its prey, it was roaring loudly.
will give branlist pls
Answer:
Roaring loudly, the lion jumped into the air to chase its prey.
Explanation:
Which is the best example
of a debatable “big idea”
for a persuasive essay
topic?
A. How many people have cell phones?
B. What should we do about cell
phones?
C. How should schools manage cell
phone use?
Answer:
A
Explanation:
what are 4 or 5 creative art ideas for class
Answer:
Use a character generator and sketch what you think that character looks like
Make a fanart of someone you like to watch, or streamer
Draw something with a single line
Print a photo out, cut it in half and try to draw the other half
Read this excerpt from We’ve Got a Job.
At that moment, nine-year-old Audrey made a decision. She would no longer just attend meetings. Somehow, she would act on her own.
Which of Audrey’s character traits is best highlighted in this excerpt?
her curiosity
her independence
her stubbornness
her cleverness
Answer: Independance
Explanation: I’ll give you a hint, acting on your own means your acting independantly Therefore when Audrey is acting on her own she is acting independently. (Sorry for bad spelling)
Answer:
b
Explanation:
Try These questions out and I’ll give you brainliest no links posted on My question or I will report you
Answer:
9.D
10.D
11.A
12.D
13.B
14.C
15.D
16.A
hope it helps you
please mark me as brainlist
classify each research source as a print or internet source
hello can someone write a sentence with the word counterintuitive in it, make it sound like a 6th grader wrote it tho
Answer:
here ya go
Explanation:
Jim decided to bake a cake and included the eggshells, which is counter intuitive because he wants a good-tasting cake.
Answer:
If you want to get better grades, going to the party instead of studying for the test tomorrow seems counterintuitive.
Explanation:
2. Identify all of the COMMON nouns in the following sentence. The kings 4 points
in Mesopotamia were expected to protect the city, provide laws, order,
and justice. *
Mesopotamia
city
laws
justice
expected
kings
Answer:
City, laws, justice, kings.
Explanation:
To answer this question, you must first know what a noun is, which is a person, place, or thing. Look through the list, and you will find out that expected is not a person, place, or thing; it is a verb.
Now you're left with Mesopotamia, city, laws, justice, and kings. A quick tip amongst nouns is that capitalized and eponymous (something that is given a name to) nouns are most probably proper nouns, which is not what we're looking for here. Cross out Mesopotamia, and you are finished.
Hope this helps! :)
need help no links on My question
Answer:
1.are
2. carry
3. gives
4. makes
5. has
"Why don't we go outside and play basketball?" suggested Kirill to his little brother Foma, who was lounging on the sofa.
What change, if any, should be made to this sentence?
OA. Change the question mark to a comma.
ОВ.
Remove the comma after Foma
C.
Add a comma after Kirill.
OD
No change is necessary.
Answer:
i think its d
Explanation:
worth 100 points!!!
Write about a time when you tried something outside of your comfort zone. Did you succeed? Why or why not?
Answer:
I once tried to go on a slip n slide which was out of my comfort zone because I was forced to, and no I did not succeed because I face planted.
what would you possess, to have power or to have authority
Answer:
To have authority.
Explanation:
While power is crucial in controlling people and getting things done, using power without authority usually requires more power to influence more people and get things done. There is also a chance to get rebelled against or people below you to switch allegiance. For example, a military coup plotter would need to exercise more force each time to let people accept him as their leader. Similarly, a street thug would need to show more power to control the street and have people fear him. The above idea makes power delicate and challenging to manage.
On the other hand, authority usually comes with power. With authority, one can easily do things that power alone can not do. With authority, one can easily command people under their control to do the right things or expected of them. Authority is equipped with lay-down rules and regulations guiding it. People are generally aware of the authority and usually comply with the authority's demands with little or no resistance.
Hence, I prefer Authority to Power because Authority is a total package. A combination of recognition and power.
Which sentence is appropriate for a formal e-mail?
v
The sentence is appropriate for a formal e-mail is option B. Healthy alternatives should be added to the vending machines to promote good health.
E - mail is a way of changing messages between humans the usage of digital devices. electronic e-mail changed into as a consequence conceived as the digital model of, or counterpart to, email, at a time while e-mail intended handiest bodily e-mail.
E-mail is a pc-based application for the alternate of messages between customers. A global community allows humans to trade messages very quickly. It the electronic equivalent of a letter, but with benefits in timeliness and versatility.
Quick for e-mail, or e mail is facts saved on a pc this is exchanged among two customers over telecommunications. extra it appears that evidently, is a message that can incorporate textual content, documents, pix, or different attachments sent through a community to a specified individual or institution of individuals.
Disclaimer:- your question is incomplete, please see below for complete question.
A. The vending machines are full of junk, and kids are forced to settle for what’s there.
B. Healthy alternatives should be added to the vending machines to promote good health.
C. When students get the munchies at the end of the day, they need healthy choices.
D. Let’s get rid of the current snack machine and replace it w/ something worthwhile!
Learn more about e-mail here:-https://brainly.com/question/24688558
#SPJ1
May have These answered again I’m Trying To check To Make sure it’s right I need help on This
Answer:
1. chilly and cool
2. walk and stroll
3. scare and frighten
4. bravery and courage
5. eat and munch
6. draw and sketch
7. thin and skinny
8. tiny and small
9. store and market
10. fall and falling
11. see and watch
12. friendly and kind
13. tasty and delicious
14. silent and quiet
15. smelly and stinky
16. smile and grin
Explanation:
hope this helped
Answer:
Question :1
1. Answer is chilly, cool.
2.Answer is walk, stroll
3. Answer is scare, fighten
4. Answer is bravery, courage
5. Answer is eat, munch
6. Answer is draw, sketch
Question :2
7. Shinny
8. small
9. Shop
10. Trip
11. Obverse
12. Kind
13. Delicious
14.quiet
15. Stinky
16. Grain
17. Ponder
18. Ruin
please help me someone
Answer:
the answer is A
Explanation:
Answer:
Orange County Pest Control Office
