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August 12th, 2017, 02:12 PM   #1
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several true\false questions in set theory

Hello and I'm so glad to join this great place.

I got a set of more than 60 true/false questions that I solved, but I am not sure regarding the following (true or false):

1) if R is a relation of set A, there exists $\displaystyle X \subseteq A$ so that $\displaystyle R = X \times X$

2) if R and S are relations over set A, R \times S is a relation over A.

3) if R is a relation over A so R × A is a relation over A × A

4) if R and S are relations over set A, then the symmetrical difference $\displaystyle R \oplus S$ is a relation over set A.

5) if $\displaystyle X \subseteq A$ AND $\displaystyle R=X \times X'$ then $\displaystyle \text{domain}(R) \cup \text{range}(R) = A$

6) if R is a relation over set A and if $\displaystyle \text{domain}(R) \cap \text{range}(R) = \varnothing$, there exists a set $\displaystyle X \subseteq A$

Thank you very much and hoping you can help me with those. I am not really sure whether they if the given claims are true or not. My due date is in a day and a half and I've spent so many hours doing it.

*I tried to use a MathJax tutorial to enter math code. I hope I did it correctly.

Thank you again and glad to join this great place!

Last edited by skipjack; August 12th, 2017 at 08:19 PM.
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August 12th, 2017, 03:01 PM   #2
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Quote:
Originally Posted by mathsnoob View Post
Hello and I'm so glad to join this great place.

I got a set of more than 60 true/false questions that I solved, but I am not sure regarding the following (true or false):

1) if R is a relation of set A, there exists $\displaystyle X \subseteq A$ so that $\displaystyle R = X \times X$

2) if R and S are relations over set A, R \times S is a relation over A.

3) if R is a relation over A so R × A is a relation over A × A

4) if R and S are relations over set A, then the symmetrical difference $\displaystyle R \oplus S$ is a relation over set A.

5) if $\displaystyle X \subseteq A$ AND $\displaystyle R=X \times X'$ then $\displaystyle \text{domain}(R) \cup \text{range}(R) = A$

6) if R is a relation over set A and if $\displaystyle \text{domain}(R) \cap \text{range}(R) = \varnothing$, there exists a set $\displaystyle X \subseteq A$

Thank you very much and hoping you can help me with those. I am not really sure whether they if the given claims are true or not. My due date is in a day and a half and I've spent so many hours doing it.

*I tried to use a MathJax tutorial to enter math code. I hope I did it correctly.

Thank you again and glad to join this great place!
I'd love to help you if you show me your thoughts on these questions. You spent so much time on it, so surely you must think something about them?

Last edited by skipjack; August 12th, 2017 at 08:19 PM.
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August 13th, 2017, 03:13 AM   #3
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Sorry for not supplying additional info. Here are my thoughts:

1) if R is a relation over set A, so I think that there can exist a set X that is well included in A so that R = X × X (because by the given details, X is a subset of A, and X × X still consists of its elements). so I think it's true.

2) if R and S are relations over set A, then R × S are relations over set A. I think it's true because if they apply the needed elements to be relations over A, then R × S must be a relation over A as well. so I think it's true.

3) if R is a relation over set A then R × A is a relation over A × A. I think it's wrong because if R is a relation, it might consist of less elements so R × A might not be relation over A × A.

4) I am really not sure regarding this one, but I think that it is a relation because as long as it's not an empty set, it is still a relation over A. so I think it's true (but really not sure).

5) if X is well included in A(subseteq) and if R = X × X', then I think that R = A at least, which means that domain(R)∪range(R)=A. So I think it's true.

6) if R is a relation over set A and if domain(R)∩range(R)=∅, there exists a set X⊆A. I think it's wrong because if there is no common variables between the domain and the range, then I am not sure there exists a subset X of A that applies R ⊆ X x X'. So I think it's wrong.

I tried to solve them, but I'm not sure and I also provided my thoughts and explanations. Can you please tell me whether I'm right or wrong? thank you in advance.

Last edited by skipjack; August 13th, 2017 at 10:33 AM.
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