I've been looking at a past paper with solutions, and I can't quite make sense of the answers given here (which is odd considering I get these questions right every time on the online practice tests), where am I going wrong with this?
Consider the set A = {a,b,c,d,e,f} and the relation defined on A X A by setting R = {(a,b),(c,d),(d,c),(b,c),(b,f),(f,a),(b,a),(b,b)}
List the elements of A X A that must be added to R to make up its reflexive closure.
Solution: {(a,a),(c,c),(d,d),(e,e),(f,f)
Does it have to include ones that weren't even included in R, such as e? If so, I guess this one is an easy one to figure out.
List the elements of A X A that must be added to R to make up its symmetric closure.
Solution: {(a,f),(f,b),(c,b),(c,a),(e,a)}
This one I can't make sense of, I'd have thought the answer would be {(c,b),(f,b),(a,f)}. I don't see where (a,c) comes from when there's no (c,a) present, and I especially don't see where the (e,a) comes from when no e is present in R.
List the elements of A X A that must be added to R to make up its transitive closure.
Solution: {(f,b),(b,d),(a,a),(c,c),(d,d),(f,f)}
I really can't make sense of this answer at all.
Would anyone be able to explain to me why these answers are true?
You need to carefully read the definitions of reflexive, symmetric, and transitive so you can apply them. Reflexive says that every element is related to itself, so you need all the pairs of the form $(x,x)$ Nothing else matters for this. I agree with your answer for symmetric. For transitive, the definition is that if we have $(x,y)$ and $(y,z)$ we must have $(x,z)$ so since we have $(c,d)$ and $(d,c)$ we must have $(c,c)$ and $(d,d)$. Similarly, since we have $(b,c)$ and $(c,d)$ we must have $(b,d)$ Can you explain why we need $(f,b)$?