several set theory true/false questions i couldn't solve (just 5 short ones)

[mathsnoob1]

hello,

i've tried to solve those for about 2 hours and i'm not sure i've done them right. i'll write the question and my attempt and please correct me if i'm wrong.
note: i should only write if the claim is true or not.

1) the function f: P(N) -> P(N) that is defined by f(x)= x - {1} for every [tax] x \in P(N)[\tax] is injective.
it seems that not every value of x has a single y value, becaue of that -{1} part, so it's false in my opinion.

2)the function f: P(N) -> P(N) that is defined by f(x)= x - {1} for every [tax] x \in P(N)[\tax] is surjective.
every point in the codomain is the value of f(x) for at least one point x in the domain.

for the next questions, the following information is given: S is a relation on Z (set of all integers) that applies [tax](n,m) \in S <-> m^2+m=n^2+n[/tax]

3)[tax](-n-1,n) \in S^2, for every n \in Z[/tax]

seems to be true since every element in S^2 is positive, since the relation S^2 = S x S.

4)S is equivalence relation over Z.

it seems to apply all the needed properties, i.e: reflexive, symmetric and transitive relations. so it's true

5)if S is a equivalnce relation, then all of its classes has the same number of variables

i think it's true because the number of classes are finite, and because in the information it is given that S only consists of (n,m). due to that, if it is an equivalnce relation and it applies its rules, then it should be reflexive, symmetric and transitive. so i think it's true.

i've tried to elaborate the answers i've given. please help me and correct me if i'm wrong. i want to improve and learn.

thank you in advance!

 

[stapel]

i've tried to solve those for about 2 hours and i'm not sure i've done them right. i'll write the question and my attempt and please correct me if i'm wrong.
note: i should only write if the claim is true or not.

1) the function f: P(N) -> P(N) that is defined by f(x)= x - {1} for every \(\displaystyle x\, \in\, P(N)\) is injective.

What is "P(N)"? Is this perhaps the power set of the naturals?

it seems that not every value of x has a single y value, becaue of that -{1} part, so it's false in my opinion.

What do you mean by "value of x", "y value", and "because of that -{1} part"?

2)the function f: P(N) -> P(N) that is defined by f(x)= x - {1} for every \(\displaystyle x\, \in\, P(N)\) is surjective.
every point in the codomain is the value of f(x) for at least one point x in the domain.

It sounds like you're mostly just restating what they're wanting you to prove. What is your proof for this statement? And how are you defining "points"?

for the next questions, the following information is given: S is a relation on Z (set of all integers) that applies \(\displaystyle (n,\,m)\, \in\, S \Longleftrightarrow \, m^2\,+\,m\, =\, n^2\, +\, n\)

3) \(\displaystyle \,(-n\,-\,1,\,n)\, \in\, S^2\, \forall \, n \, \in\, Z\)

seems to be true since every element in S^2 is positive, since the relation S^2 = S x S.

Are you sure that this question has been copied correctly? In what sense can a two-coordinate point be "positive"? On what basis are you concluding "positivity" from the fact that the space S^2 is a set of points? Or are you thinking that S^2 means the square of some number or something...?

4)S is equivalence relation over Z.

it seems to apply all the needed properties, i.e: reflexive, symmetric and transitive relations. so it's true

What do you mean by "it seems to" and "apply"? What is your proof that the relation S definitely does (or does not) possess these properties?

5)if S is a equivalnce relation, then all of its classes has the same number of variables

i think it's true because the number of classes are finite, and because in the information it is given that S only consists of (n,m). due to that, if it is an equivalnce relation and it applies its rules, then it should be reflexive, symmetric and transitive. so i think it's true.

This isn't asking you if this is an equivalence relation. It is asking you to say something about the sizes of its classes. What is your proof of this?

Yes, the answers to these questions may be required to be nothing more than "true" or "false", but your justification of your answers (such as on the test, where you have to show your work) needs to contain proofs. What have you done in this regard for any of these questions?

Please be complete. Thank you!