I'm trying to solve these problems:
Decide if the following relations are partial order:
a) Relation ≼ is relation on the set $\mathbb{R}^{3}$, where $(a_{1}, a_{2}, a_{3})$ ≼ $(b_{1}, b_{2}, b_{3})$ if $(a_{1}, a_{2}, a_{3})$ = $(b_{1}, b_{2}, b_{3})$ or there is a maximum of one $i\in$ {$1, 2, 3$} such that $a_{i}\geq b_{i}$.
I know that partial order is reflexive, antisymmetric, and transitive.
Reflexivity: $(a_{1}, a_{2}, a_{3})$ ≼ $(a_{1}, a_{2}, a_{3})$ if $(a_{1}, a_{2}, a_{3})$ = $(a_{1}, a_{2}, a_{3})$ - it is obviously true.
Transitivity: $(a_{1}, a_{2}, a_{3})$ ≼ $(b_{1}, b_{2}, b_{3})$ $\wedge$ $(b_{1}, b_{2}, b_{3})$ ≼ $(c_{1}, c_{2}, c_{3})$ if
$(a_{1}, a_{2}, a_{3})$ = $(b_{1}, b_{2}, b_{3})$ $\wedge$ $(b_{1}, b_{2}, b_{3})$ = $(c_{1}, c_{2}, c_{3})$ → $(a_{1}, a_{2}, a_{3})$ = $(c_{1}, c_{2}, c_{3})$.Antisymmetry: I am sure I have to use the second statement (there is a maximum of one $i\in$ {$1, 2, 3$} such that $a_{i}\geq b_{i}$) but I don't know how.
b) Relation ≼ is relation on the set $\mathbb{N}$, where $a$ ≼ $b$ if $a$ = $b$ or $a$ $\leq$ $10b$.
- Reflexivity: $a$ ≼ $a$ just when $a$ = $a$ - it is obviously true.
- Transitivity: $a$ ≼ $b$ $\wedge$ $b$ ≼ $c$ if $a$ = $b$ $\wedge$ $b$ = $c$ → $a$ = $c$.
- Antisymmetry: $a$ ≼ $b$ $\wedge$ $b$ ≼ $a$ $\Rightarrow$ $a$ = $b$ → $a$ $\leq$ $10b$ $\wedge$ $b$ $\leq$ $10a$ $\Rightarrow$ $a$ $\leq$ $10b$ $\leq$ $100a$ $\Rightarrow$ $a$ $\leq$ $100a$.
Is it so, or I made a mistake somewhere?
I'm going to use lte to denote the relation in each problem.
Both of your transitivity proofs are incorrect. You need to consider the second portion of the relation described. For example, (0,0,0) $\preceq$ (-1,-1,1) since only the third coordinate of (-1,-1,1) is greater than or equal to the third coordinate of (0,0,0). Now consider all of the (a, b, c) such that (-1,-1,0) $\preceq$ (a, b, c) and check if (0,0,0,) $\preceq$ (a, b, c). What you have so far in your proof shows that when (a, b, c) = (-1,-1,1) you have (0,0,0,) $\preceq$ (a, b, c). What about all of the other (-1,-1,1) $\preceq$ (a, b, c)? You need to do the same for transitivity in part b.
For part B, your antisymmetry proof is incorrect. Try playing around with a = 1. If 1 $\preceq$ b and b $\preceq$ 1, does b = 1? In other words if 1 is less than or equal to 10*b and b is less than or equal to 10*1, is b = 1? Then you need to expand your argument to general a and b?