I dont know what I know or dont know, so I dont know what to study,
Like the stuff in the beginning, I already know all that shit, but I am still doing fuck bad in class for some fucking reason its just that the teacher doesnt like my format.
I dont have the answers to the problems at my disposal, so I never know if I am right or not... how am I right or how am I wrong.
Maybe it is because I missed the phrase, “inserting parentheticremarks to explain the rationale behind each step”.
A convergent sequence is bounded. (Begin by writing this as a conditional involving a variable.)
-what is a convergent sequence?
-Is there a way to prove this?
-How would a proof for this look like?
Write the following with quantifiers. (I want to see the comprehensive quantifiers... well use as much as possible like you know, dont simplify it. And tell me what you mean by each symbol and shit)
-Some number raised to the third power is negative.
-The sine of a number is always between -1 and 1, inclusive.
Is the next one as easy as I think it is?
Express the following laws symbolically:
a) the commutative law of multiplication
b) the associative law of addition
c) the distributive law
So I asked the prof about De Morgans law, and after I talked to him I realized he gave me answers that contradicted... so, what is De Morgan’s Law? Is it negation of a statement or is it stating the same statement by using negation of the opposite statement?
Whats with the GIANT conjunction/disjunction symbols? I think it has to do with sigma notation, what do they represent/mean and what are they used for?
Can you explain why?: the negation of “everybody is stupid” is NOT “everybody is not stupid”
How would you say this in English (how would the order of what you refer to change?)?
-(Universal Quanti.)x, (Exis. Quanti.)y, (Uni. Quanti.)z (If x<y then x<z<y)
[this statement is false by the way]
-What were the easy proofs like, you know the ones from highschool geometry, is there anything from those that I can apply to these proofs? Why or why not?
-What is the main goal and procedure when proving that something exists, and when proving that something is unique?
- What do the brackets [] in Zn mean? Does it have anything to do with equivalence classes? If so, how so?
- In Zn, Im suppose to prove uniqueness for [x] when [a], [b] are in Zn , [x] is a subset of Zn, and [a] + [x] = [b] ::::
Premises: [a],[b] and are in Zn and [a] + [x] = [b]
[x] = [b] – [a]
Since [a] and [b] can only be between 0... (n-1) then [x] only has one value within Zn
- I thought this may not be what he wants so I tried doing it differently but I dont know if the following is right:
Suppose [x] < [x’]
[x] = [b] – [a] and [x’] = [b] – [a]
[b] – [a] = [b] – [a]
So [x] = [x’]
So it is not true that [x] < [x’]
I think this must not be right since 4 = 22 or (-2)2
And although 4 = 4
22 does not = (-2)2
Am I reasoning correctly? Why am I right or wrong?
I was also thinking of trying each [] number individually, would that have worked?
If Z4 can [0] = [5] = [10]?
How the hell am I suppose to do this?
In Z12 find all elements [x] such that [x]n = [0] n = positive integer
The integers a and b are relatively prime if (a, b) = 1.
What does it mean for something tobe relatively prime? What can it do?
In (a,p)=1 ... 1<or=a<or=p-1.
Why -1?
1)What makes two numbers relatively prime?
I started off by listing each number and listing each of their relative primes.
1:1, 2, 3, 4, 5.....(infinite)
2:3, 5, 7, 9... (all odds... or 2n+1)
3: 4, 7... (3n+1) and (3n+2)
4: 5, 7, 9, ... (4n+1) or might as well (2n+1) again
5: 6, 7, 8, 9, 11, 12, 13, 14, ... (5n+1), (5n+2), (5n+3), (5n+4)
6: 7, 11, 13, 17, 19, ... (6n+1), (6n-1)
2)Should you be able to confirm the properties with one round? Example:
Relative primes of 6 include... check (012345)... DONE
2a)yes... generalize and it turns out to be true.
0=6=12=..., 1=7=13=..., 2=8=14=..., 3=9=15=..., 4=10=16=..., 5=11=17=...
(6n), (6n+1), (6n+2), (6n+3), (6n+4), (6n+5), [correspondingly]
Also notice that:
(6n)=(6n), (6n+1)=(6n-5), (6n+2)=(6n-4), (6n+3)=(6n-3), (6n+4)=(6n-2), (6n+5)=(6n-1)
Notice! All even numbers do not have even numbers that are relative prime, I guess it is because the greatest common factor between even numbers will be sure to at least be 2.
Its like even numbers are their own species and dont need odd numbers to function...2 is the 1 of even numbers dude!!...(3 is the 1 of multiples of 3 & 4 is the 1 of multiples of 4...hmm)
Definition of even number- multiple of 2
Anyways:
1:...
2:... & (no evens)
3:...
4...& (no evens)
5:...
6:...& (no evens)
I got it!
If you take a look at each number’s prime factorization you will see...
Whatever prime factorizations two numbers share, those are some of their common factors.
1:1, 1
2:1, 2
3:1, 3
4:1, 2, 4
5:1, 5
6:1, 2, 3, ... (must also include the multiples of the factors... 4, 6)
7:1, 7
8:1, 2,...4, 6, 8
9:1, 3
10:1, 2, 5
-And notice all even numbers, of course, have a common factor of 2; which is why two even numbers can never be relatively prime.
+odd numbers do not carry the prime factor of 2
-Prime numbers do not have many factors so any number would work, unless it is a multiple of that number of course.
-This is why any number (n) is always relatively prime to its adjacents (n+1) or (n-1).... idk
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