My note have define the binary relation as follow but I find out I cannot understand the meaning of the symbols.
let R be a binary relation on A
what is a binary relation? My note only said "A relation R on A x B is called a binary relation form a to B. If A = B then we simply say R is a binary relation on A". if this is only relation on A where is the x,y,z (below) come form.
R is reflexive iff (x,x) for all x in A $\forall x [(x,x) \in R]$ is true
what is (x,x) mean? and what is "is true" mean?
R is symmetric iff $\forall x \forall y [(x, y) \in R \Rightarrow (y,x) \in R]$
what is the different between (x,y) and (y,x)? and why using$ \Rightarrow$ insteat of $\rightarrow$?
R is antisymmetric iff $\forall x \forall y [((x,y)\in R \land (y,x)\in R)\Rightarrow (x=y)]$
what is antisymmetric mean? Why is the defination so different form the symmetric?
R is transitive iff $\forall x \forall y \forall z[((x,y)\in R \land (y,z)\in R)\Rightarrow (x,z)\in R]$
why the result is only (x,z)? where is the y?