I'm currently taking an intro discrete math course, and I'm having some trouble understanding the rules of reflexivity, symmetry, and transitivity. The book isn't making a lot of sense to me, and my professors examples haven't been very clear. I have two problems in which I'm not confident in my answers.
1) There is a set S = {1, 2, 3, 4}, and a relation R on S, defined by R = {(1,1),(3,4),(4,2),(4,1),(3,2)}. I need to determine if R is reflexive, symmetric, and transitive.
A relation is reflexive if, and only if, xRx for all elements x in S, correct? So there is only one tuple where x relates to itself, so R is not reflexive
A relation is symmetric if, and only if, if xRy then yRx for all (x,y) pairs in S, correct? In this relation, there are no instances where x relates to y and y relates to x, so R is not symmetric
A relation is transitive if, and only if, xRy and yRz then xRz for all x,y,z pairs in S, correct? This relation has one instance of transitivity, where 3R4 and 4R2 then 3R2, but transitivity doesn't hold true for all pairs (pairs including 1), so R is not transitive?
2) There is a set S = {all real numbers}, and a relation R defined by: R is the set of all ordered pairs (x,y) such that x and y are real numbers, 1 < x < 2 AND x = y. I need to determine if R is reflexive, symmetric, and transitive.
Since, by definition of R, x = y wouldn't that mean that every (x,y) tuple that would be part of R would contain an x and y such that x = y, so the elements would relate to themselves? So they would be reflexive. But this relation is also limited to 1 < x <2. So this relation obviously doesn't hold all possible ordered pairs possible in the set of real numbers. So I would say that R is not reflexive. Is that correct reasoning?
By definition of R, I know that x = y so y = x, so the proposition if xRy then yRx would be true in this case. So I would think that R is symmetric, but I don't know how the stipulation 1 < x <2 plays into this. Also, This relation won't have every ordered pair in S, so would that mean it's not symmetric?
You could prove transitivity with the transitive law if x=y and y=z then x=z, but again I don't see how it would apply to all ordered pairs or satisfy the 1 < x < 2, so I'd say not transitive
tl;dr:
I'm having problems proving reflexivity, symmetry, and transitivity of the following:
1) set S = {1,2,3,4} relation R = {(1,1),(3,4),(4,2),(4,1),(3,2)}
2) set S = {all real numbers} relation R = {(x,y)|1 < x < 2 and x=y}