sorry for the dumbest question ever, but i want to understand total order in an intuitive way,
this is the defition of total order:
i) If $a ≤ b$ and $b ≤ a$ then $a = b$ (antisymmetry);
ii) If $a ≤ b$ and $b ≤ c$ then $a ≤ c$ (transitivity);
iii) $a ≤ b$ or $b ≤ a$ (totality).
totality means that any pair of the total ordered pair is mutually comparable. i dont understand what they mean under comparable, what is the defition of comparability? i can also compare the elements of partial order, where is problem? why is partial order not not mutually comparable?
can someone explain me please in simple words 🙁
Best Answer
Two distinct elements are called "comparable" when one of them is greater than the other. This is the definition of "comparable". When you have a partially ordered set, some pairs of elements can be not comparable. i.e. you can have two elements $x$ and $y$ such that $x\leqslant y$ is false and $y \leqslant x$ is also false.
For example, consider the set $\mathbb{R}^2$ and a partial order defined like this: $$ (x_1,x_2) \leqslant (y_1,y_2) \quad\textrm{iff}\quad x_1\leqslant y_1 \,\textrm{and}\,x_2 \leqslant y_2. $$ With this partial order, elements $(0,0)$ and $(1,2)$ of $\mathbb{R^2}$ are comparable, because $(0,0)\leqslant(1,2)$. But elements $(0,1)$ and $(1,0)$ are not comparable, because both statements "$(0,1)\leqslant(1,0)$" and "$(1,0)\leqslant(0,1)$" are false.