For puzzle 4-

NOTE: leq denotes less than equal to.

1) Set of all finite words (using the English alphabet) with dictionary ordering- x leq y if the word x occurs before y or at the same position as y in a dictionary (here a finite word is just a finite sequence of letters).

This is a preorder which is also a (skeletal) partial order.

2) Consider a non-empty finite set where every element is related to every other element including itself. If the set has more than 1 element then this is a preorder which is not a (skeletal) partial order .

One could think of the finite set as a family of people with leq thought of as "related to". So, everyone in a family is related to each other (we're assuming a person is related to him or herself), So, person x is related to person y and person y is related to person x, but this doesn't mean that person x is the same as person y.

NOTE: leq denotes less than equal to.

1) Set of all finite words (using the English alphabet) with dictionary ordering- x leq y if the word x occurs before y or at the same position as y in a dictionary (here a finite word is just a finite sequence of letters).

This is a preorder which is also a (skeletal) partial order.

2) Consider a non-empty finite set where every element is related to every other element including itself. If the set has more than 1 element then this is a preorder which is not a (skeletal) partial order .

One could think of the finite set as a family of people with leq thought of as "related to". So, everyone in a family is related to each other (we're assuming a person is related to him or herself), So, person x is related to person y and person y is related to person x, but this doesn't mean that person x is the same as person y.