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August 12th, 2017, 02:12 PM  #1 
Newbie Joined: Aug 2017 From: mathmatistan Posts: 9 Thanks: 0  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. 
August 12th, 2017, 03:01 PM  #2  
Senior Member Joined: Oct 2009 Posts: 555 Thanks: 179  Quote:
Last edited by skipjack; August 12th, 2017 at 08:19 PM.  
August 13th, 2017, 03:13 AM  #3 
Newbie Joined: Aug 2017 From: mathmatistan Posts: 9 Thanks: 0 
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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