I'm trying to determine if the following are symmetric, reflexive, transitive, equivalence for all-natural numbers but am struggling because they aren't in set notation.
Examples of confusing notation:
- xRy: |x-y|=2
I'm assuming in analysing these I have to keep the statements true. I understand, a relation is symmetric if I can plug in x into a function, get y and then plug in y (in the same function) to get x.
So for the example would it be symmetric if I can plug (a into x, b in y) and (b into y and a into x) and both statements be true or would it be reflexive as long as they evaluate to the same thing?
Now for understanding if its transitive. My thinking is if I can plug (a into x, b into y) then (b into x, c into y) and finaly (c into x, a in y) Once again does it have to be true for all values of a,b,c or does it simply have to false or true for all statement?
The notation you see here can be expanded to $$R = \{(x,y) : |x-y| = 2\}$$ Now you should see that this relation is symmetric but neither reflexive nor transitive.
Generally the notations $xRy: P(x,y)$ and $R = \{(x,y) : P(x,y)\}$, where $P$ is a predicate, are equivalent. Sometimes you also see $xRy :\Leftrightarrow P(x,y)$ (read the $:\Leftrightarrow$ as "is defined equivalent to")