Suppose that $A,B$ are distinct lattice points in $\mathbb Z^n$. We say that $A$ dominates $B$ if all the components of $A-B$ are non-negative. Given positive integers $a_1,a_2,\dots ,a_n$, let $S$ be a set of lattice points in the integer lattice $L=[0,a_1]\times[0,a_2]\times\dots\times...

This is the set $S=\{ 0,1,4,6,7,8,9\}$ under the order defined by divisibility. I know we have to find antichains and chains, and that the maximum number of partitions of $S$ into chains should equal the cardinality of the antichains, but i don't understand union of chains and how that would fit...

Many books say that Hall's marriage theorem is equivalent to Dilworth's theorem. Some use König's theorem to show that, but many just don't prove it at all. Is there any simple approach to understand and later prove why this applies using set theory (without graph theory), relating the chains an...

I'm looking for some interesting examples or theorems about partial orders to show to my students, can be also some fundamental theorem such as an analogue of this theorem on equivalence relations. Any help is welcome Thanks

This is the set $S=\{ 0,1,4,6,7,8,9\}$ under the order defined by divisibility. I know we have to find antichains and chains, and that the maximum number of partitions of $S$ into chains should equal the cardinality of the antichains, but i don't understand union of chains and how that would fit...

Many books say that Hall's marriage theorem is equivalent to Dilworth's theorem. Some use König's theorem to show that, but many just don't prove it at all. Is there any simple approach to understand and later prove why this applies using set theory (without graph theory), relating the chains an...

I'm looking for some interesting examples or theorems about partial orders to show to my students, can be also some fundamental theorem such as an analogue of this theorem on equivalence relations. Any help is welcome Thanks

I was trying to solve this question. If $f ◦f$ is differentiable, then $f ◦f ◦f$ is differentiable. While trying to find the counterexample. I come across the Dirichlet function. $f(x) = \begin{cases} 1 & x \in \mathbb{Q},\\ 0 & x \not\in\mathbb{Q}\end{cases}$. $f ◦f=1$. I landed up on my own def...